LIMIT SOLUTIONS (on estensions of input-output for
unbounded controls)
Franco Rampazzo, Maria Soledad Aronna
To cite this version:
Franco Rampazzo, Maria Soledad Aronna. LIMIT SOLUTIONS (on estensions of input-output
for unbounded controls). NETCO 2014 - New Trends in Optimal Control, Jun 2014, Tours,
France. <hal-01024721>
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https://hal.inria.fr/hal-01024721
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LIMIT SOLUTIONS
(on estensions of input-output for unbounded
controls)
Franco Rampazzo,
University of Padova,
Padova, Italy
joint work with Maria S. Aronna (IMPA)
NetCo Conference
New trends in Optimal Control
June 23-27, 2014
Tours, France
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
1 / New
37
1
HEURISTICS
2
”LIMIT” SOLUTIONS
Existing notions of solutions
Proposed definition of Limit Solution
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
2 / New
37
HEURISTICS
1
HEURISTICS
2
”LIMIT” SOLUTIONS
Existing notions of solutions
Proposed definition of Limit Solution
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
3 / New
37
HEURISTICS
TWO MAIN QUESTIONS:
(A) Find a NOTION OF SOLUTION x for
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
t ∈ [a, b]
α=1
such that:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
4 / New
37
HEURISTICS
TWO MAIN QUESTIONS:
(A) Find a NOTION OF SOLUTION x for
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
t ∈ [a, b]
α=1
such that:
i) x is L1 and is defined for L1 inputs u
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
4 / New
37
HEURISTICS
TWO MAIN QUESTIONS:
(A) Find a NOTION OF SOLUTION x for
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
t ∈ [a, b]
α=1
such that:
i) x is L1 and is defined for L1 inputs u
Here L1 denotes the set of integrable maps defined everywhere
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
4 / New
37
HEURISTICS
TWO MAIN QUESTIONS:
(A) Find a NOTION OF SOLUTION x for
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
t ∈ [a, b]
α=1
such that:
i) x is L1 and is defined for L1 inputs u
Here L1 denotes the set of integrable maps defined everywhere
ii) x subsumes former concepts of solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
4 / New
37
HEURISTICS
TWO MAIN QUESTIONS:
(A) Find a NOTION OF SOLUTION x for
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
t ∈ [a, b]
α=1
such that:
i) x is L1 and is defined for L1 inputs u
Here L1 denotes the set of integrable maps defined everywhere
ii) x subsumes former concepts of solution.
We shall call this solution
LIMIT SOLUTION
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
4 / New
37
HEURISTICS
TWO MAIN QUESTIONS:
(A) Find a NOTION OF SOLUTION x for
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
t ∈ [a, b]
α=1
such that:
i) x is L1 and is defined for L1 inputs u
Here L1 denotes the set of integrable maps defined everywhere
ii) x subsumes former concepts of solution.
We shall call this solution
LIMIT SOLUTION
(B) Minimum problems with end-point constraints x(b) ∈ S as
PROPER EXTENSIONS of regular problems (u ∈ AC )?
(Aronna’s talk)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
4 / New
37
HEURISTICS
APPLICATIONS of impulsive systems:
Spiking models of synaptic behaviour
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
5 / New
37
HEURISTICS
APPLICATIONS of impulsive systems:
Spiking models of synaptic behaviour
Mechanical systems using some coordinates as controls
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
5 / New
37
HEURISTICS
APPLICATIONS of impulsive systems:
Spiking models of synaptic behaviour
Mechanical systems using some coordinates as controls
In general, coupled fast-slow dynamics
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
5 / New
37
HEURISTICS
Underlying thought:
We can accept a notion of discontinuous (=impulsive) trajectory
only if
it is, in some sense to be made precise, the limit of faster and faster
trajectories
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
6 / New
37
”LIMIT” SOLUTIONS
Outline
1
HEURISTICS
2
”LIMIT” SOLUTIONS
Existing notions of solutions
Proposed definition of Limit Solution
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
7 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
8 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
A ”TRIVIAL” BUT IMPORTANT CASE
ẋ = u̇
For this eqution one would like to have
x(t) = u(t) + x(0)
∀t ∈ [0, T ]
(1)
as a solution, which is obviously true as soon as x, u ∈ AC (i.e.
absolutely continuous).
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
8 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
A ”TRIVIAL” BUT IMPORTANT CASE
ẋ = u̇
For this eqution one would like to have
x(t) = u(t) + x(0)
∀t ∈ [0, T ]
(1)
as a solution, which is obviously true as soon as x, u ∈ AC (i.e.
absolutely continuous).
Another idea could be a distributional approach: BUT
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
8 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
A ”TRIVIAL” BUT IMPORTANT CASE
ẋ = u̇
For this eqution one would like to have
x(t) = u(t) + x(0)
∀t ∈ [0, T ]
(1)
as a solution, which is obviously true as soon as x, u ∈ AC (i.e.
absolutely continuous).
Another idea could be a distributional approach: BUT
1) does not give pointwise information
2) it is ”wrong” in the general nonlinear case.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
8 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
A ”TRIVIAL” BUT IMPORTANT CASE
ẋ = u̇
For this eqution one would like to have
x(t) = u(t) + x(0)
∀t ∈ [0, T ]
(1)
as a solution, which is obviously true as soon as x, u ∈ AC (i.e.
absolutely continuous).
Another idea could be a distributional approach: BUT
1) does not give pointwise information
2) it is ”wrong” in the general nonlinear case.
Can we take (1) as a definition for u, x ∈ L1 ?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
8 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
EXISTING NOTIONS OF SOLUTION FOR
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
9 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
EXISTING NOTIONS OF SOLUTION FOR
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
There are at least TWO cases for which a
”good” notion of solution already does exist:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
9 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
EXISTING NOTIONS OF SOLUTION FOR
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
There are at least TWO cases for which a
”good” notion of solution already does exist:
the commutative case
[gα , gβ ] = 0
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
9 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
EXISTING NOTIONS OF SOLUTION FOR
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α
α=1
There are at least TWO cases for which a
”good” notion of solution already does exist:
the commutative case
[gα , gβ ] = 0
the non commutative case
[gα , gβ ] 6= 0
with the controls u(·) having bounded variation
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
9 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case =the ”trivial” case:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case =the ”trivial” case:
1 Due to [g , g ] = 0, by multiple flow-box theorem there exists a
α β
(global) coordinates’change
(x, u) → (ξ, z) = ξ(x, u), u
such that the system becomes trivial:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case =the ”trivial” case:
1 Due to [g , g ] = 0, by multiple flow-box theorem there exists a
α β
(global) coordinates’change
(x, u) → (ξ, z) = ξ(x, u), u
such that the system becomes trivial:
ξ˙ = F (t, ξ, z, v )
ż = u̇
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case =the ”trivial” case:
1 Due to [g , g ] = 0, by multiple flow-box theorem there exists a
α β
(global) coordinates’change
(x, u) → (ξ, z) = ξ(x, u), u
such that the system becomes trivial:
ξ˙ = F (t, ξ, z, v )
ż = u̇
2
set z(t) := u(t) and define the solution x(·) by using the inverse
change of coordinates:
x(t) = x ξ(t), z(t)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case =the ”trivial” case:
1 Due to [g , g ] = 0, by multiple flow-box theorem there exists a
α β
(global) coordinates’change
(x, u) → (ξ, z) = ξ(x, u), u
such that the system becomes trivial:
ξ˙ = F (t, ξ, z, v )
ż = u̇
2
set z(t) := u(t) and define the solution x(·) by using the inverse
change of coordinates:
x(t) = x ξ(t), z(t)
Notice: One has continuity of u → x with respect to L1 topologies on
both u and x.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
P
Existing notions of solutions
gα (x, u)u̇α
The commutative case =the ”trivial” case:
1 Due to [g , g ] = 0, by multiple flow-box theorem there exists a
α β
(global) coordinates’change
(x, u) → (ξ, z) = ξ(x, u), u
such that the system becomes trivial:
ξ˙ = F (t, ξ, z, v )
ż = u̇
2
set z(t) := u(t) and define the solution x(·) by using the inverse
change of coordinates:
x(t) = x ξ(t), z(t)
Notice: One has continuity of u → x with respect to L1 topologies on
both u and x. Actually, stronger continuity properties are verified
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
10 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
References include
A. Bressan and F. Rampazzo. Impulsive control systems with commutative
vector fields. J. Optim. Theory Appl., 71, p.67-83, (1991).
A.V. Sarychev. Nonlinear systems with impulsive and generalized function
controls,vol. 9 of Progr. Systems Control Theory, p. 244-257, (1991).
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
11 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Existing notions of solutions
gα (x, u)u̇α ,
The noncommutative case
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
12 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Existing notions of solutions
gα (x, u)u̇α ,
The noncommutative case =non ”trivializable”:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
12 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Existing notions of solutions
gα (x, u)u̇α ,
The noncommutative case =non ”trivializable”:
1
For regular (=absolutely continuous) controls u, one can
reparameterize time t(s) = ϕ0 (s) and set ϕ(s) := u ◦ ϕ0 = u(t(s)):
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
12 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Existing notions of solutions
gα (x, u)u̇α ,
The noncommutative case =non ”trivializable”:
1
For regular (=absolutely continuous) controls u, one can
reparameterize time t(s) = ϕ0 (s) and set ϕ(s) := u ◦ ϕ0 = u(t(s)):
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
2
for BV(=bounded variation) controls u, let (ϕ0 , ϕ) be a graph
completions of u .
Namely:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
12 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Existing notions of solutions
gα (x, u)u̇α ,
The noncommutative case =non ”trivializable”:
1
For regular (=absolutely continuous) controls u, one can
reparameterize time t(s) = ϕ0 (s) and set ϕ(s) := u ◦ ϕ0 = u(t(s)):
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
2
for BV(=bounded variation) controls u, let (ϕ0 , ϕ) be a graph
completions of u .
Namely: one bridges the jumps of u and parameterize them on
s-subintervals where time t(s)(= ϕ0 (s)) is constant.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
12 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
t → x(t) := y ◦ ϕ−1
0 (t)
is called the graph-completion solution corresponting to the graph
completion (ϕ0 , ϕ) of u. It is set-valued on a countable subset of [a, b].
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
13 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
t → x(t) := y ◦ ϕ−1
0 (t)
is called the graph-completion solution corresponting to the graph
completion (ϕ0 , ϕ) of u. It is set-valued on a countable subset of [a, b].
single-valued version:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
13 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
t → x(t) := y ◦ ϕ−1
0 (t)
is called the graph-completion solution corresponting to the graph
completion (ϕ0 , ϕ) of u. It is set-valued on a countable subset of [a, b].
single-valued version: If σ : [0, T ] → [0, 1] is a Clock, i.e.
σ(t) ∈ (ϕ0 , ϕ)← (t, u(t)), we say that
t → x := y ◦ σ(t)
is a single-valued graph-completion solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
13 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
14 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
15 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
16 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
17 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
18 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
t 0 (s) = ϕ00 (s)
y 0 (s) = f (ϕ0 , y , ϕ, v ◦ ϕ0 )ϕ00 (s) +
m
X
gα (y , u)ϕ0α (s)
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
19 / New
37
”LIMIT” SOLUTIONS
Existing notions of solutions
An incomplete list of authors who have investigated this subject:
Bressan
Bressan- Rampazzo
Bressan-Mazzola
Briani-Zidani
Pereira-Vinte
Miller
Motta-Rampazzo
Camilli-Falcone
Motta-Sartori
Silva-Vinter.
Zabic-Wolenski
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
20 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Some requirements should be met:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Some requirements should be met:
consistency with the Carathéodory notion of solution for u ∈ AC ;
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Some requirements should be met:
consistency with the Carathéodory notion of solution for u ∈ AC ;
x single-valued at each t;
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Some requirements should be met:
consistency with the Carathéodory notion of solution for u ∈ AC ;
x single-valued at each t;
existence of an output
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Some requirements should be met:
consistency with the Carathéodory notion of solution for u ∈ AC ;
x single-valued at each t;
existence of an output (and possibly uniqueness) results
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
A unified notion of solution x: LIMIT SOLUTION
M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. (accepted
for publ. on JDE)
Some requirements should be met:
consistency with the Carathéodory notion of solution for u ∈ AC ;
x single-valued at each t;
existence of an output (and possibly uniqueness) results
former definitions of solution for impulsive systems should be
subsumed by this extended notion
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
21 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α ,
x(a) = x̄
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
22 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α ,
x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 )
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
22 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α ,
x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 )
Definition
A map x : [a, b] → Rn is a LIMIT SOLUTION if, for every
τ ∈ [a, b], there exists a sequence of absolutely continuous controls
(ukτ ) such that
|(xkτ , ukτ )(τ ) − (x, u)(τ )| + k(xkτ , ukτ ) − (x, u)k1 → 0,
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
22 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α ,
x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 )
Definition
A map x : [a, b] → Rn is a LIMIT SOLUTION if, for every
τ ∈ [a, b], there exists a sequence of absolutely continuous controls
(ukτ ) such that
|(xkτ , ukτ )(τ ) − (x, u)(τ )| + k(xkτ , ukτ ) − (x, u)k1 → 0,
SIMPLE LIMIT SOLUTION: if (ukτ ) can be chosen
independently of τ , i.e. (ukτ ) = (uk ).
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
22 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α ,
x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 )
Definition
A map x : [a, b] → Rn is a LIMIT SOLUTION if, for every
τ ∈ [a, b], there exists a sequence of absolutely continuous controls
(ukτ ) such that
|(xkτ , ukτ )(τ ) − (x, u)(τ )| + k(xkτ , ukτ ) − (x, u)k1 → 0,
SIMPLE LIMIT SOLUTION: if (ukτ ) can be chosen
independently of τ , i.e. (ukτ ) = (uk ).
BV-SIMPLE LIMIT SOLUTION if the approximating inputs uk
have equibounded variation.
(xkτ is the Carathéodory solution corresponding to (ukτ , v ))
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
22 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
23 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Theorem 1.
Existence and uniqueness For every control u ∈ L1 (and every
v ∈ L1 ) there exists a unique limit solution of
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
23 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Theorem 1.
Existence and uniqueness For every control u ∈ L1 (and every
v ∈ L1 ) there exists a unique limit solution of
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Continuous dependence: for every τ ∈ [a, b] one has
|x1h(τ ) − x2 (τ )| + kx1 − x2 k1 ≤
i
M |x̄1 − x̄2 | + |u1 (a) − u2 (a)| + |u1 (t) − u2 (t)| + ku1 − u2 k1 .
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
23 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Theorem 1.
Existence and uniqueness For every control u ∈ L1 (and every
v ∈ L1 ) there exists a unique limit solution of
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Continuous dependence: for every τ ∈ [a, b] one has
|x1h(τ ) − x2 (τ )| + kx1 − x2 k1 ≤
i
M |x̄1 − x̄2 | + |u1 (a) − u2 (a)| + |u1 (t) − u2 (t)| + ku1 − u2 k1 .
moreover: one has continuous dependence w.r. to the standard control v (·)
in L1 norm
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
23 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
24 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Proposition 2.The limit solution coincides with the solution previously
given via change of coordinates.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
24 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Proposition 2.The limit solution coincides with the solution previously
given via change of coordinates.
Remarks:
(1) Proposition 2 was proved in Bressan-Rampazzo(1991) when f
independent of v .
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
24 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE COMMUTATIVE CASE, [gα , gβ ] = 0
Proposition 2.The limit solution coincides with the solution previously
given via change of coordinates.
Remarks:
(1) Proposition 2 was proved in Bressan-Rampazzo(1991) when f
independent of v .
(2) Proposition 2 gives an idea for the general definition of solution
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
24 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Example
ẋ = xv + x u̇,
x(0) = x̄,
on the interval [0, 1], with v (t) := χ[0,1/2[
Consider the L1 control
(−1)k+1 , for t ∈ [1 − k1 , 1 −
u(t) :=
0,
for t = 1.
The limit solution x is given by

x̄e t ,
for


 x̄e 1/2 e −2 , for
x(t) :=
x̄e 1/2 ,
for



x̄e −1/2 , for
t
t
t
t
∈ [0, 21 [,
S
∈ ∞
k=1 [1 −
S∞
∈ k=1 [1 −
= 1.
1
k+1 [,
k ∈ N,
1
1
2k , 1 − 2k+1 [,
1
1
2k+1 , 1 − 2k+2 [,
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
25 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Example
ẋ = xv + x u̇,
x(0) = x̄,
on the interval [0, 1], with v (t) := χ[0,1/2[
Consider the L1 control
(−1)k+1 , for t ∈ [1 − k1 , 1 −
u(t) :=
0,
for t = 1.
The limit solution x is given by

x̄e t ,
for


 x̄e 1/2 e −2 , for
x(t) :=
x̄e 1/2 ,
for



x̄e −1/2 , for
t
t
t
t
∈ [0, 21 [,
S
∈ ∞
k=1 [1 −
S∞
∈ k=1 [1 −
= 1.
1
k+1 [,
k ∈ N,
1
1
2k , 1 − 2k+1 [,
1
1
2k+1 , 1 − 2k+2 [,
Notice that both u and x have infinitely many discontinuities, unbounded
variation, and are defined everywhere.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
25 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE GENERIC, NON COMMUTATIVE, CASE
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
26 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE GENERIC, NON COMMUTATIVE, CASE
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 ).
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
26 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE GENERIC, NON COMMUTATIVE, CASE
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 ).
Existence of limit solutions?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
26 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE GENERIC, NON COMMUTATIVE, CASE
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 ).
Existence of limit solutions?
Existence of simple limit solutions, possibly BV?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
26 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE GENERIC, NON COMMUTATIVE, CASE
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 ).
Existence of limit solutions?
Existence of simple limit solutions, possibly BV?
Uniqueness?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
26 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
THE GENERIC, NON COMMUTATIVE, CASE
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
u ∈ L1 ([a, b]; U), (and v ∈ L1 ).
Existence of limit solutions?
Existence of simple limit solutions, possibly BV?
Uniqueness?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
26 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Proposed definition of Limit Solution
gα (x, u)u̇α ,
x(a) = x̄
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
27 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Proposed definition of Limit Solution
gα (x, u)u̇α ,
x(a) = x̄
Let us focus on BV-simple limit solutions (for u ∈ BV ). Do they have
something to to with graph-completion solutions?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
27 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Proposed definition of Limit Solution
gα (x, u)u̇α ,
x(a) = x̄
Let us focus on BV-simple limit solutions (for u ∈ BV ). Do they have
something to to with graph-completion solutions?
Theorem
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
27 / New
37
”LIMIT” SOLUTIONS
ẋ = f (x, u, v ) +
X
Proposed definition of Limit Solution
gα (x, u)u̇α ,
x(a) = x̄
Let us focus on BV-simple limit solutions (for u ∈ BV ). Do they have
something to to with graph-completion solutions?
Theorem
x a single-valued graph completion solution
~
w

x is a BV-simple limit solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
27 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Theorem
x a single-valued graph completion solution
~
w

x is a BV-simple limit solution.
Main ingredients of the proof:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
28 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Theorem
x a single-valued graph completion solution
~
w

x is a BV-simple limit solution.
Main ingredients of the proof:
w
w
 (more or less known): pointwise density for increasing maps plus
reparameterizations;
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
28 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Theorem
x a single-valued graph completion solution
~
w

x is a BV-simple limit solution.
Main ingredients of the proof:
w
w
 (more or less known): pointwise density for increasing maps plus
reparameterizations;
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
28 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Theorem
x a single-valued graph completion solution
~
w

x is a BV-simple limit solution.
Main ingredients of the proof:
w
w
 (more or less known): pointwise density for increasing maps plus
reparameterizations;
~
w
w (new) : compactness, by Helly’s and Ascoli-Arzelà’s theorem, plus
ad hoc approximation tecqniques.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
NetCo controls)
Conference
28 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Observe preliminarly that the question is not obvious even for the trivial
equation
ẋ = u̇
x(a) = 0
Indeed:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
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estensions
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Aronna (IMPA)
for unbounded
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Conference
29 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Observe preliminarly that the question is not obvious even for the trivial
equation
ẋ = u̇
x(a) = 0
Indeed:
claiming (as we do) that x(t) ≡ u(t), t ∈ [a, b], would mean that the BV
map u : [a, b] → IRm can be approximated pointwise by a sequence of
absolutely continuous maps un with Var (un ) ≤ L.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
29 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Observe preliminarly that the question is not obvious even for the trivial
equation
ẋ = u̇
x(a) = 0
Indeed:
claiming (as we do) that x(t) ≡ u(t), t ∈ [a, b], would mean that the BV
map u : [a, b] → IRm can be approximated pointwise by a sequence of
absolutely continuous maps un with Var (un ) ≤ L.
(This is not straightforward: consider e.g. a BV map with a dense set of
discontinuities)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
29 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
30 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Theorem
Let U have the Whitney property.
For any control pair
(u, v ) ∈ BV([a, b]; U) × L1 ([a, b]; V )
there exists a BV-simple limit solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
NetCo controls)
Conference
30 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Theorem
Let U have the Whitney property.
For any control pair
(u, v ) ∈ BV([a, b]; U) × L1 ([a, b]; V )
there exists a BV-simple limit solution.
(An arc-wise connected set U has the Whitney property if d(x, y ) ≤ M|x − y |,
where d is the geodesic distance.)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
NetCo controls)
Conference
30 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
EXISTENCE of BV-SIMPLE LIMIT SOLUTIONS for
m
X
ẋ = f (x, u, v ) +
gα (x, u)u̇α , x(a) = x̄
α=1
Theorem
Let U have the Whitney property.
For any control pair
(u, v ) ∈ BV([a, b]; U) × L1 ([a, b]; V )
there exists a BV-simple limit solution.
(An arc-wise connected set U has the Whitney property if d(x, y ) ≤ M|x − y |,
where d is the geodesic distance.)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
30 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Let u ∈ AC .
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Let u ∈ AC .
Clearly the Carathéodory solution xC of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄
α=1
is a (BV-uniform) limit solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Let u ∈ AC .
Clearly the Carathéodory solution xC of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄
α=1
is a (BV-uniform) limit solution.
Question:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Let u ∈ AC .
Clearly the Carathéodory solution xC of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄
α=1
is a (BV-uniform) limit solution.
Question:
Is xC the ONLY limit solution?
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
NetCo controls)
Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Let u ∈ AC .
Clearly the Carathéodory solution xC of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄
α=1
is a (BV-uniform) limit solution.
Question:
Is xC the ONLY limit solution?
NO.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
Let u ∈ AC .
Clearly the Carathéodory solution xC of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄
α=1
is a (BV-uniform) limit solution.
Question:
Is xC the ONLY limit solution?
NO.
For instance
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
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estensions
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Aronna (IMPA)
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Conference
31 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
On the other hand, the input
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
On the other hand, the input
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
x(0) = 0.
g2 (x) := (0, 1, −x1 ),
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
On the other hand, the input
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
Since uk (t) → (0, 0),
xk (t) → (0, 0, −t)t , uniformly, the map
x̂(t) := (0, 0, −t)t
is a (simple) limit solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
x(0) = 0.
g2 (x) := (0, 1, −x1 ),
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
On the other hand, the input
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
Since uk (t) → (0, 0),
xk (t) → (0, 0, −t)t , uniformly, the map
x̂(t) := (0, 0, −t)t
is a (simple) limit solution. In particular x̂ 6= xC
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
NetCo controls)
Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
x(0) = 0.
g2 (x) := (0, 1, −x1 ),
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
On the other hand, the input
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
Since uk (t) → (0, 0),
xk (t) → (0, 0, −t)t , uniformly, the map
x̂(t) := (0, 0, −t)t
is a (simple) limit solution. In particular x̂ 6= xC
Notice that Var (uk ) → +∞
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Counterexample to uniqueness
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
x(0) = 0.
g2 (x) := (0, 1, −x1 ),
so [g1 , g2 ] = (0, 0, −2).
Of course the Carathéodory solution corresponding to u ≡ (0, 0) is
xC (t) ≡ (0, 0, 0)
On the other hand, the input
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
Since uk (t) → (0, 0),
xk (t) → (0, 0, −t)t , uniformly, the map
x̂(t) := (0, 0, −t)t
is a (simple) limit solution. In particular x̂ 6= xC
Notice that Var (uk ) → +∞
BUT...
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
32 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
u ∈ AC
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
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Conference
33 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
u ∈ AC
THEOREM.
Let x̂ ∈ AC be a BV-uniform limit solution of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄.
α=1
Then x̂ = xC
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
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Conference
33 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
u ∈ AC
THEOREM.
Let x̂ ∈ AC be a BV-uniform limit solution of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄.
α=1
Then x̂ = xC
Remark : By the previous example, the fact that x̂ is a smooth simple
limit solution does not imply that x is a Carathéodory solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
33 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
CONSISTENCY with Carathéodory solutions xC
u ∈ AC
THEOREM.
Let x̂ ∈ AC be a BV-uniform limit solution of
ẋ = f (x, u, v ) +
m
X
gα (x, u)u̇α ,
x(a) = x̄.
α=1
Then x̂ = xC
Remark : By the previous example, the fact that x̂ is a smooth simple
limit solution does not imply that x is a Carathéodory solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
33 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
SOME PROBLEMS:
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
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Aronna (IMPA)
for unbounded
NetCo controls)
Conference
34 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
SOME PROBLEMS:
Proper extension for minimum problems (next talk by Aronna)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
34 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
SOME PROBLEMS:
Proper extension for minimum problems (next talk by Aronna)
Classify other classes of solutions with unbounded variation.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
34 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
SOME PROBLEMS:
Proper extension for minimum problems (next talk by Aronna)
Classify other classes of solutions with unbounded variation.
Necessary conditions for minimum problems, Hamilton-Jacobi
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
34 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
SOME PROBLEMS:
Proper extension for minimum problems (next talk by Aronna)
Classify other classes of solutions with unbounded variation.
Necessary conditions for minimum problems, Hamilton-Jacobi
Compactness, existence of minima
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
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Conference
34 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Many thanks for your patience
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
35 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
36 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Carathéodory solution corresponding to u ≡ (0, 0) is : xC (t) ≡ (0, 0, 0)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Carathéodory solution corresponding to u ≡ (0, 0) is : xC (t) ≡ (0, 0, 0)
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt)
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Carathéodory solution corresponding to u ≡ (0, 0) is : xC (t) ≡ (0, 0, 0)
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
xk (t)tox̂(t) := (0, 0, −t)t
so x̂ is a (simple) limit solution.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Carathéodory solution corresponding to u ≡ (0, 0) is : xC (t) ≡ (0, 0, 0)
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
xk (t)tox̂(t) := (0, 0, −t)t
so x̂ is a (simple) limit solution. In particular x̂ 6= xC
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
g2 (x) := (0, 1, −x1 ),
x(0) = 0.
so [g1 , g2 ] = (0, 0, −2).
Carathéodory solution corresponding to u ≡ (0, 0) is : xC (t) ≡ (0, 0, 0)
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
xk (t)tox̂(t) := (0, 0, −t)t
so x̂ is a (simple) limit solution. In particular x̂ 6= xC
. R1
Notice that Var (uk ) = 0 |u̇k |dt → +∞
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37
”LIMIT” SOLUTIONS
Proposed definition of Limit Solution
The BUT... stuff
ẋ = g1 (x)u̇1 + g2 (x)u̇2 ,
g1 (x) := (1, 0, x2 ),
x(0) = 0.
g2 (x) := (0, 1, −x1 ),
so [g1 , g2 ] = (0, 0, −2).
Carathéodory solution corresponding to u ≡ (0, 0) is : xC (t) ≡ (0, 0, 0)
uk (t) := (k −1/2 cos kt − 1, k −1/2 sin kt) generates the trajectory
xk (t) = (k −1/2 cos kt − 1, k −1/2 sin kt, −t + k −1 sin kt)t .
xk (t)tox̂(t) := (0, 0, −t)t
so x̂ is a (simple) limit solution. In particular x̂ 6= xC
. R1
Notice that Var (uk ) = 0 |u̇k |dt → +∞ BUT... the iterated integral
Z
1
|u̇k2 uk1 − u̇k1 uk2 |dt
0
IS BOUNDED as k goes to ∞.
Franco Rampazzo, University of Padova, Padova,
LIMIT SOLUTIONS
Italy joint work
(on with
estensions
MariaofS.input-output
Aronna (IMPA)
for unbounded
NetCo controls)
Conference
37 / New
37