LOCAL MINIMUM PRINCIPLE FOR OPTIMAL CONTROL
PROBLEMS SUBJECT TO INDEX ONE
DIFFERENTIAL-ALGEBRAIC EQUATIONS
MATTHIAS GERDTS∗
Abstract. Necessary conditions in terms of a local minimum principle are derived for optimal
control problems subject to index-1 differential-algebraic equations, pure state constraints and mixed
control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which frequently arise in practical applications. The local minimum
principle is based on necessary optimality conditions for general infinite optimization problems. The
special structure of the optimal control problem under consideration is exploited and allows to obtain
more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz like
constraint qualification for the optimal control problem ensures the regularity of a local minimum.
Key words. optimal control, necessary conditions, local minimum principle, index one differentialalgebraic equation system, state constraints
AMS subject classifications. 49K15, 34A09
1. Introduction. Optimal control problems subject to ordinary differential equations have a wide range of applications in different disciplines like engineering sciences, chemical engineering, and economics. Necessary conditions known as ‘Maximum principles’ or ‘Minimum principles’ have been investigated intensively since the
1950’s. Early proofs of the maximum principle are given by Pontryagin et al. [28]
and Hestenes [9]. Necessary conditions with pure state constraints are discussed in,
e.g., Jacobsen et al. [12], Girsanov [6], Knobloch [14], Maurer [22, 23], Ioffe and Tihomirov [11], and Kreindler [15]. Neustadt [27] and Zeidan [31] discuss optimal control
problems with mixed control-state constraints. Hartl et al. [7] provide a survey on
maximum principles for optimal control problems with state constraints including an
extensive list of references. Necessary conditions for variational problems, i.e. smooth
optimal control problems, are developed in Bryson and Ho [1]. Second order necessary
conditions and sufficient conditions are stated in Zeidan[31]. Sufficient conditions are
also presented in Maurer [24] and Malanowski [21].
Necessary conditions are not only interesting from a theoretical point of view, but
also provide the basis of the so-called indirect approach for solving optimal control
problems numerically. In this approach the minimum principle is exploited and usually
leads to a multi-point boundary value problem, which is solved numerically by, e.g.,
the multiple shooting method. Nevertheless, even for the direct approach, which
is based on a suitable discretization of the optimal control problem, the minimum
principle is very important for the post-optimal approximation of adjoints. In this
context, the multipliers resulting from the formulation of the necessary Fritz-John
conditions for the finite dimensional discretized optimal control problem have to be
related to the multipliers of the original infinite dimensional optimal control problem
in an appropriate way. It is evident that this requires the knowledge of necessary
conditions for the optimal control problem.
We will extend the results for optimal control problems with ordinary differential
equations to problems involving differential-algebraic equations (DAE’s). Differentialalgebraic equations are composite systems of differential equations and algebraic equa∗ Fachbereich Mathematik (OA), Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany ([email protected]).
1
2
M. GERDTS
tions. Particularly, we will discuss DAE systems of type
ẋ(t) = f (t, x(t), y(t), u(t))
0ny = g(t, x(t), y(t), u(t))
a.e. in [t0 , tf ],
a.e. in [t0 , tf ].
(1.1)
(1.2)
Herein, x(t) ∈ Rnx is referred to as differential variable and y(t) ∈ Rny as algebraic
variable. Correspondingly, (1.1) is called differential equation and (1.2) algebraic
equation. The variable u is a control variable, which is an external function and
allows to control the system in an appropriate way. In this paper we will restrict the
discussion to so-called semi-explicit index-1 DAE systems. Semi-explicit index-1 DAE
systems are characterized by the postulation that the matrix gy0 (t, x(t), y(t), u(t)) is
non-singular. Semi-explicit Index-1 systems often occur in process system engineering,
cf. Hinsberger [10], but also in vehicle simulation, cf. Gerdts [5], and many other fields
of applications.
Necessary conditions for optimal control problems subject to index-1 DAE systems without state constraints and without mixed control-state constraints can be
found in de Pinho and Vinter [2]. Implicit control systems are discussed in Devdariani and Ledyaev [3]. In our representation, we followed to a large extend the
approaches in Kirsch et al. [13] and Machielsen [20].
We investigate nonlinear optimal control problems of the subsequent form. Let
[t0 , tf ] ⊂ R be a non-empty and bounded interval with fixed time points t0 , tf and
U ⊆ Rnu a closed and convex set with non-empty interior. Let
ϕ : Rnx × Rnx → R,
f0 : [t0 , tf ] × Rnx × Rny × Rnu
f : [t0 , tf ] × Rnx × Rny × Rnu
g : [t0 , tf ] × Rnx × Rny × Rnu
ψ : R n x × R nx → R n ψ ,
c : [t0 , tf ] × Rnx × Rny × Rnu
s : [t0 , tf ] × Rnx → Rns
→ R,
→ Rnx ,
→ Rny ,
→ Rnc ,
be mappings. For n ∈ N, the space L∞ ([t0 , tf ], Rn ) consists of all measurable functions
h : [t0 , tf ] → Rn with
khk∞ := ess supkh(t)k < ∞,
t0 ≤t≤tf
where k · k denotes the Euclidian norm on Rn . With this norm, L∞ ([t0 , tf ], Rn )
becomes a Banach space. The space W 1,∞ ([t0 , tf ], Rn ) consists of all absolutely continuous functions h : [t0 , tf ] → Rn with
khk1,∞ := max{khk∞ , kh0 k∞ } < ∞,
where h0 denotes the first derivative of h. The space W 1,∞ ([a, b], Rn ) endowed with
the norm k · k1,∞ is a Banach space. We will denote the null element of a general
Banach space X by ΘX or, if no confusion is possible, simply by Θ. We will use 0n
if X = Rn and 0 if X = R.
We consider
Problem 1.1 (DAE optimal control problem).
Find an absolutely continuous differential variable x ∈ W 1,∞ ([t0 , tf ], Rnx ), an essentially bounded algebraic variable y ∈ L∞ ([t0 , tf ], Rny ), and an essentially bounded
LOCAL MINIMUM PRINCIPLE
control variable u ∈ L∞ ([t0 , tf ], Rnu ) such that the objective function
Z tf
ϕ(x(t0 ), x(tf )) +
f0 (t, x(t), y(t), u(t))dt
3
(1.3)
t0
is minimized subject to the semi-explicit differential algebraic equation (1.1)-(1.2), the
boundary conditions
ψ(x(t0 ), x(tf )) = 0nψ ,
(1.4)
the mixed control-state constraints
c(t, x(t), y(t), u(t)) ≤ 0nc
a.e. in [t0 , tf ],
(1.5)
the pure state constraints
s(t, x(t)) ≤ 0ns
in [t0 , tf ],
(1.6)
and the set constraints
u(t) ∈ U
a.e. in [t0 , tf ].
(1.7)
Some definitions and terminologies are in order. (x, y, u) ∈ W 1,∞ ([t0 , tf ], Rnx ) ×
L ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu ) is called admissible or feasible for the optimal
control problem 1.1, if the constraints (1.1)-(1.7) are fulfilled. An admissible pair
∞
(x̂, ŷ, û) ∈ W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu )
is called a weak local minimum of Problem 1.1, if there exists ε > 0 such that
Z tf
ϕ(x̂(t0 ), x̂(tf )) +
f0 (t, x̂(t), ŷ(t), û(t))dt
t0
tf
Z
≤ ϕ(x(t0 ), x(tf )) +
f0 (t, x(t), y(t), u(t))dt
t0
holds for all admissible (x, y, u) with kx− x̂k1,∞ < ε, ky − ŷk∞ < ε, and ku− ûk∞ < ε.
An admissible pair
(x̂, ŷ, û) ∈ W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu )
is called a strong local minimum of Problem 1.1, if there exists ε > 0 such that
Z tf
ϕ(x̂(t0 ), x̂(tf )) +
f0 (t, x̂(t), ŷ(t), û(t))dt
t0
tf
Z
≤ ϕ(x(t0 ), x(tf )) +
f0 (t, x(t), y(t), u(t))dt
t0
holds for all admissible (x, y, u) with kx − x̂k∞ < ε.
Notice, that strong local minima are also weak local minima. The converse is
not true. Strong local minima are minimal w.r.t. a larger class of algebraic variables
and controls. Weak local minima are only optimal w.r.t. all algebraic variables and
controls in a L∞ -neighborhood.
4
M. GERDTS
The paper is organized as follows. In Section 2 necessary conditions for general
infinite optimization problems known from the literature are summarized. These
necessary conditions are formulated for the optimal control problem in Section 3.
Explicit representations of the multipliers involved in the necessary conditions are
derived in Section 4. These explicit representations are exploited in order to state
the main result of the paper in Section 5 – the local minimum principle for optimal
control problems of type (1.1). In addition, some important special cases are discussed.
Section 6 summarizes a constraint qualification which ensures the regularity of a local
minimum. Finally, some concluding remarks close the paper.
2. Necessary conditions for infinite optimization problems. We summarize results for optimization problems in Banach spaces. Let (X, k·kX ), (Y, k·kY ), (Z, k·
kZ ) be Banach spaces, f : X → R a functional and g : X → Y , h : X → Z operators.
The topological dual spaces of X, Y, Z are denoted by X ∗ , Y ∗ , Z ∗ , respectively. Let
S ⊆ X be a closed convex set with non-empty interior and K ⊆ Y a closed convex
cone with vertex at zero. K + := {y ∗ ∈ Y ∗ | y ∗ (y) ≥ 0 for all y ∈ K} is referred to as
the positive dual cone of K.
Consider the optimization problem
min f (x)
s.t. x ∈ S, g(x) ∈ K, h(x) = ΘZ .
(2.1)
Theorem 2.1. Let x̂ be a local minimum of the optimization problem (2.1). Let
f : X → R and g : X → Y be Fréchet-differentiable at x̂, h : X → Z continuously
Fréchet-differentiable at x̂, K ⊆ Y a closed convex cone with vertex at zero, and S a
closed convex set with non-empty interior. Furthermore, assume that
(i) int(K) 6= ∅ and
(ii) im(h0 (x̂)) is not a proper dense subspace in Z.
Then there exist nontrivial multipliers (0, ΘY ∗ , ΘZ ∗ ) 6= (l0 , η ∗ , λ∗ ) ∈ R × Y ∗ × Z ∗ such
that
l0 ≥ 0,
η∗ ∈ K + ,
∗
η (g(x̂)) = 0,
0
∗ 0
∗ 0
l0 f (x̂)(d) − η (g (x̂)(d)) − λ (h (x̂)(d)) ≥ 0, ∀d ∈ S − {x̂}.
(2.2)
(2.3)
(2.4)
(2.5)
Proof. Kurcyusz [16], Th. 3.1, Cor. 4.2, Lempio [17]
Conditions which ensure that the multiplier l0 in Theorem 2.1 is not zero are
called regularity conditions or constraint qualifications. In this case, without loss of
generality l0 can be normalized to one. The subsequent Mangasarian-Fromowitz like
condition is such a regularity condition.
Corollary 2.2. Let the assumptions of Theorem 2.1 be satisfied. In addition,
let the following assumptions be fulfilled:
(i) h0 (x̂) is surjective.
(ii) There exists some dˆ ∈ int(S − {x̂}) with
ˆ = ΘZ ,
h0 (x̂)(d)
0
ˆ ∈ int(K − {g(x̂)}).
g (x̂)(d)
(2.6)
(2.7)
LOCAL MINIMUM PRINCIPLE
5
Then, the assertions of Theorem 2.1 hold with l0 = 1.
Proof. Follows from Theorem 2.1 by contradiction, if it is assumed, that the
assertions in Theorem 2.1 hold with l0 = 0.
Remark 2.3. The above regularity condition of Mangasarian-Fromowitz is equivalent to the famous regularity condition of Robinson [29]
(ΘY , ΘZ ) ∈ int{(g 0 (x̂)(s − x̂) − k + g(x̂), h0 (x̂)(s − x̂)) | s ∈ S, k ∈ K}.
(2.8)
Robinson [29] postulated this condition in the context of stability analysis of generalized
inequalities. Zowe and Kurcyusz [32] used it to show l0 = 1.
3. Abstract formulation of the optimal control problem. In order to obtain necessary conditions for a weak local minimum, we reformulate the general problem 1.1 as an optimization problem in appropriate Banach spaces. We follow the
approaches of Kirsch et al. [13] and Machielsen [20].
For notational convenience throughout this chapter we will use the abbreviations
ϕ0x0 := ϕ0x0 (x̂(t0 ), x̂(tf )),
fx0 [t] := fx0 (t, x̂(t), ŷ(t), û(t)),
0
0
0
and in a similar way ϕ0xf , f0,x
[t], f0,y
[t], f0,u
[t], c0x [t], c0y [t], c0u [t], s0x [t], fy0 [t],
0
0
0
0
0
0
fu [t], gx [t], gy [t], gu [t], ψx0 , ψxf for the respective derivatives.
The variables (x, y, u) are taken as elements from the Banach space
X := W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu )
endowed with the norm
k(x, y, u)kX := max{kxk1,∞ , kyk∞ , kuk∞ }.
The objective function F : X → R is given by
Z tf
F (x, y, u) := ϕ(x(t0 ), x(tf )) +
f0 (t, x(t), y(t), u(t))dt.
t0
If ϕ and f0 are continuous w.r.t. all arguments and continuously differentiable w.r.t.
to x, y, and u, then F is Fréchet-differentiable at (x̂, ŷ, û) with
F 0 (x̂, ŷ, û)(δx, δy, δu) = ϕ0x0 δx(t0 ) + ϕ0xf δx(tf )
Z tf
0
0
0
+
f0,x
[t]δx(t) + f0,y
[t]δy(t) + f0,u
[t]δu(t)dt,
t0
cf. Kirsch et al. [13], p. 94. Now we collect the equality constraints. The space
Z := L∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × Rnψ
endowed with the norm
k(z1 , z2 , z3 )kZ := max{kz1 k∞ , kz2 k∞ , kz3 k2 }
is a Banach space. The equality constraints of the optimal control problem are given
by
H(x, y, u) = ΘZ ,
6
M. GERDTS
where H = (H1 , H2 , H3 ) : X → Z with
H1 (x, y, u) = f (·, x(·), y(·), u(·)) − ẋ(·),
H2 (x, y, u) = g(·, x(·), y(·), u(·)),
H3 (x, y, u) = −ψ(x(t0 ), x(tf )).
If f , g, and ψ are continuous w.r.t. all arguments and continuously differentiable
w.r.t. to x, y, and u, then H is continuously Fréchet-differentiable at (x̂, ŷ, û) with
 0

H1 (x̂, ŷ, û)(δx, δy, δu)
H 0 (x̂, ŷ, û)(δx, δy, δu) =  H20 (x̂, ŷ, û)(δx, δy, δu) 
H30 (x̂, ŷ, û)(δx, δy, δu)
and
˙
H10 (x̂, ŷ, û)(δx, δy, δu) = fx0 [·]δx(·) + fy0 [·]δy(·) + fu0 [·]δu(·) − δx(·),
0
0
0
0
H2 (x̂, ŷ, û)(δx, δy, δu) = gx [·]δx(·) + gy [·]δy(·) + gu [·]δu(·),
H30 (x̂, ŷ, û)(δx, δy, δu) = −ψx0 0 δx(t0 ) − ψx0 f δx(tf ),
cf. Kirsch et al. [13], p. 95. Now we collect the inequality constraints. The space
Y := L∞ ([t0 , tf ], Rnc ) × C([t0 , tf ], Rns )
endowed with the norm
k(y1 , y2 )kY := max{ky1 k∞ , ky2 k∞ }
is a Banach space. The inequality constraints of the optimal control problem are given
by
G(x, y, u) ∈ K,
where G = (G1 , G2 ) : X → Y and
G1 (x, y, u) = −c(·, x(·), y(·), u(·)),
G2 (x, y, u) = −s(·, x(·)).
The cone K := K1 × K2 ⊆ Y is defined by
K1 := {z ∈ L∞ ([t0 , tf ], Rnc ) | z(t) ≥ 0nc a.e. in [t0 , tf ]},
K2 := {z ∈ C([t0 , tf ], Rns ) | z(t) ≥ 0ns in [t0 , tf ]}.
If c and s are continuous w.r.t. all arguments and continuously differentiable w.r.t.
to x, y, u, then G is continuously Fréchet-differentiable at (x̂, ŷ, û) with
0
G1 (x̂, ŷ, û)(δx, δy, δu)
G0 (x̂, ŷ, û)(δx, δy, δu) =
G02 (x̂, ŷ, û)(δx, δy, δu)
and
G01 (x̂, ŷ, û)(δx, δy, δu) = −c0x [·]δx(·) − c0y [·]δy(·) − c0u [·]δu(·),
G02 (x̂, ŷ, û)(δx, δy, δu) = −s0x [·]δx(·).
7
LOCAL MINIMUM PRINCIPLE
Finally, the set S ⊆ X is given by
S := W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × Uad ,
where
Uad := {u ∈ L∞ ([t0 , tf ], Rnu ) | u(t) ∈ U a.e. in [t0 , tf ]}.
Summarizing, the optimal control problem 1.1 is equivalent with
Problem 3.1. Find (x, y, u) ∈ X such that F (x, y, u) is minimized subject to the
constraints
G(x, y, u) ∈ K,
H(x, y, u) = ΘZ ,
(x, y, u) ∈ S.
We intend to apply the necessary conditions in Theorem 2.1 to Problem 3.1. In
order to show the surjectivity of the equality constraints, we need the following results
about linear differential algebraic equations and boundary value problems.
Lemma 3.2. Consider the linear DAE
ẋ(t) = A1 (t)x(t) + B1 (t)y(t) + h1 (t),
0ny = A2 (t)x(t) + B2 (t)y(t) + h2 (t),
(3.1)
(3.2)
where A1 (t) ∈ Rnx ×nx , A2 (t) ∈ Rny ×nx , B1 (t) ∈ Rnx ×ny , B2 (t) ∈ Rny ×ny are time dependent matrix functions with entries from L∞ ([t0 , tf ], R) and h1 ∈ L∞ ([t0 , tf ], Rnx ),
h2 ∈ L∞ ([t0 , tf ], Rny ).
Let B2 (t) be non-singular almost everywhere in [t0 , tf ] and let B2 (t)−1 be essentially bounded. Define
A(t) := A1 (t) − B1 (t)B2 (t)−1 A2 (t),
h(t) := h1 (t) − B1 (t)B2 (t)−1 h2 (t).
(a) The initial value problem given by (3.1)-(3.2) together with the initial value
x(t0 ) = x0 has a unique solution x ∈ W 1,∞ ([t0 , tf ], Rnx ), y ∈ L∞ ([t0 , tf ], Rny )
for every x0 ∈ Rnx , every h1 ∈ L∞ ([t0 , tf ], Rnx ), and every h2 ∈ L∞ ([t0 , tf ], Rny ).
The solution is given by
Z t
−1
x(t) = Φ(t) x0 +
Φ (τ )h(τ )dτ ,
in [t0 , tf ],
(3.3)
t0
y(t) = −B2 (t)−1 (h2 (t) + A2 (t)x(t))
a.e. in [t0 , tf ],
(3.4)
where the fundamental system Φ(t) ∈ Rnx ×nx is the unique solution of
Φ̇(t) = A(t)Φ(t),
Φ(t0 ) = Inx .
(3.5)
(b) Let a vector b ∈ Rr and matrices C0 , Cf ∈ Rr×nx be given, such that
rank (C0 Φ(t0 ) + Cf Φ(tf )) = r
holds for the fundamental solution Φ from (a). Then, the boundary value
problem given by (3.1)-(3.2) together with the boundary condition
C0 x(t0 ) + Cf x(tf ) = b
has a solution for every b ∈ Rr .
(3.6)
8
M. GERDTS
Proof. see Appendix A
Part (a) of Lemma 3.2 enables us to formulate necessary conditions for the optimal
control problem 1.1 if we impose
Assumption 3.3. Let the index of the DAE (1.1)-(1.2) be one, i.e. let
gy0 [t] := gy0 (t, x̂(t), ŷ(t), û(t))
be non-singular a.e. in [t0 , tf ] and let gy0 [t]−1 be essentially bounded.
Remark 3.4. Notice, that by Assumption 3.3 the algebraic equation (1.2) can
be solved for ŷ and ŷ can be expressed as a function of t, x̂, û. By exploitation of
the minimum principle known for ordinary differential equations this approach would
lead to a second version of the proof of the upcoming minimum principle for DAE
systems. However, we do not follow this approach, since we think it is important to
give a direct proof in order to gain more insights in the underlying functional analytic
tools. Moreover, in the future more general DAE systems will be considered, where
this simple transformation is not possible anymore.
Furthermore, we need the following abstract result.
Lemma 3.5. Let X, Y be Banach spaces. Let T : X → Y × Rn be defined by
T (x) = (T1 (x), T1 (x)), where T1 : X → Y is a linear, continuous, and surjective
operator and T2 : X → Rn is linear and continuous. Then im(T ) = T (X) is closed in
Y × Rn .
Proof. see Appendix A
Theorem 3.6 (Necessary Conditions). Let the following assumptions be fulfilled
for the optimal control problem 1.1.
(i) Let the functions ϕ, f0 , f, g, ψ, c, s be continuous w.r.t. all arguments and
continuously differentiable w.r.t. x, y, and u.
(ii) Let U ⊆ Rnu be a closed and convex set with non-empty interior.
(iii) Let (x̂, ŷ, û) ∈ X be a weak local minimum of the optimal control problem.
(iv) Let Assumption 3.3 be valid.
Then there exist non-trivial multipliers l0 ∈ R, η ∗ ∈ Y ∗ , λ∗ ∈ Z ∗ with
l0 ≥ 0,
η∗ ∈ K + ,
∗
η (G(x̂, ŷ, û)) = 0,
0
l0 F (x̂, ŷ, û)(x − x̂, y − ŷ, u − û)
∗
−η (G0 (x̂, ŷ, û)(x − x̂, y − ŷ, u − û))
−λ∗ (H 0 (x̂, ŷ, û)(x − x̂, y − ŷ, u − û)) ≥ 0 ∀(x, y, u) ∈ S.
(3.7)
(3.8)
(3.9)
(3.10)
Proof. We show, that all assumptions of Theorem 2.1 are satisfied. Observe, that
K is a closed convex cone with vertex at zero, int(K) is non-empty, the functions F
and G are Fréchet-differentiable, and H is continuously Fréchet-differentiable due to
the smoothness assumptions. Since U is supposed to be closed and convex with nonempty interior, the set S is closed and convex with non-empty interior. It remains to
show that im(H 0 (x̂, ŷ, û)) is not a proper dense subset of Z. According to part (a) of
Lemma 3.2 the operator (H10 (x̂, ŷ, û), H20 (x̂, ŷ, û)) is continuous, linear and surjective.
Thus, we can apply Lemma 3.5, which yields that the image of H 0 (x̂, ŷ, û) is closed in
Z and hence im(H 0 (x̂, ŷ, û)) is not a proper dense subset in Z. Hence, all assumptions
of Theorem 2.1 are satisfied and Theorem 2.1 yields (3.7)-(3.10).
9
LOCAL MINIMUM PRINCIPLE
Notice, that the multipliers are elements of the following dual spaces:
∗
∗
η ∗ := (η1∗ , η2∗ ) ∈ Y ∗ = (L∞ ([t0 , tf ], Rnc )) × (C([t0 , tf ], Rns )) ,
∗
∗
λ∗ := (λ∗1 , λ∗2 , σ) ∈ Z ∗ = (L∞ ([t0 , tf ], Rnx )) × (L∞ ([t0 , tf ], Rny )) × Rnψ .
In the sequel, the special structure of the functions F , G, and H in (3.10) is exploited.
Furthermore, the fact, that the variational inequality (3.10) holds for all (x, y, u) ∈ S,
i.e. for all x ∈ W 1,∞ ([t0 , tf ], Rnx ), all y ∈ L∞ ([t0 , tf ], Rny ), and all u ∈ Uad , is used
to derive three single variational equalities and inequalities, respectively. Evaluation
of the Fréchet-derivatives in (3.10) and setting y = ŷ, u = û yields the variational
equality
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) +
Z
tf
0
l0 f0,x
[t]δx(t)dt
t0
˙
+η1∗ (c0x [·]δx(·)) + η2∗ (s0x [·]δx(·)) + λ∗1 (δx(·)
− fx0 [·]δx(·)) − λ∗2 (gx0 [·]δx(·))
= 0,
(3.11)
which holds for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). Similarly, we find
Z
tf
0
[t]δy(t)dt + η1∗ (c0y [·]δy(·)) − λ∗1 (fy0 [·]δy(·)) − λ∗2 (gy0 [·]δy(·))
l0 f0,y
= 0
(3.12)
t0
for all δy ∈ L∞ ([t0 , tf ], Rny ) and
Z
tf
0
l0 f0,u
[t]δu(t)dt + η1∗ (c0u [·]δu(·)) − λ∗1 (fu0 [·]δu(·)) − λ∗2 (gu0 [·]δu(·)) ≥ 0
t0
(3.13)
for all δu ∈ Uad − {û}.
Recall, that the multiplier η2∗ is an element of the dual space of the space of continuous functions. Hence, by Riesz’ representation theorem the functional η2∗ admits
the representation
η2∗ (h)
=
ns Z
X
i=1
tf
hi (t)dµi (t)
(3.14)
t0
for every continuous function h ∈ C([t0 , tf ], Rns ). Herein, µi , i = 1, . . . , ns are
functions of bounded variation. To make the representation unique, we choose µi ,
i = 1, . . . , ns from the space N BV ([t0 , tf ], R), i.e. the space of normalized functions of bounded variation which are continuous from the right in (t0 , tf ) and satisfy
µi (t0 ) = 0 for i = 1, . . . , ns , cf. Luenberger [19] and Natanson [26].
Replacing η2∗ in (3.11) by (3.14) and rearranging terms we obtain
Z tf
0
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) +
l0 f0,x
[t]δx(t)dt
t0
Z
n
s
tf
X
+
s0i,x [t]δx(t)dµi (t) + η1∗ (c0x [·]δx(·))
i=1
t0
˙
+λ∗1 (δx(·)
− fx0 [·]δx(·)) − λ∗2 (gx0 [·]δx(·))
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ).
= 0
(3.15)
10
M. GERDTS
4. Representation of Multipliers. In this section, useful representations of
the functionals λ∗1 λ∗2 , and η1∗ are derived and properties of Stieltjes integrals will
be exploited extensively. These results about Stieltjes integrals can be found in the
books of Natanson [26] and Widder [30].
According to Lemma 3.2 the initial value problem
˙
δx(t)
= fx0 [t]δx(t) + fy0 [t]δy(t) + h1 (t),
0ny = gx0 [t]δx(t) + gy0 [t]δy(t) + h2 (t),
δx(t0 ) = 0nx ,
(4.1)
(4.2)
has a solution for every h1 ∈ L∞ ([t0 , tf ], Rnx ) and every h2 ∈ L∞ ([t0 , tf ], Rny ). According to (3.3) and (3.4) in Lemma 3.2 the solution is given by
Z t
Φ−1 (τ )h(τ )dτ,
(4.3)
δx(t) = Φ(t)
t0
−1
0
gy [t]
(gx0 [t]δx(t) + h2 (t))
Z t
−1 0
Φ−1 (τ )h(τ )dτ + h2 (t) ,
gx [t]Φ(t)
= − gy0 [t]
δy(t) = −
(4.4)
t0
where
−1
h(t) = h1 (t) − fy0 [t] gy0 [t]
h2 (t)
and Φ is the solution of Φ̇(t) = A(t)Φ(t), Φ(t0 ) = Inx with
−1 0
A(t) = fx0 [t] − fy0 [t] gy0 [t]
gx [t].
Now, let h1 ∈ L∞ ([t0 , tf ], Rnx ) and h2 ∈ L∞ ([t0 , tf ], Rny ) be arbitrary. Adding
equations (3.12) and (3.15) and exploiting the linearity of the functionals leads to
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf )
Z tf
ns Z tf
X
0
0
(4.5)
+
l0 f0,x
[t]δx(t) + l0 f0,y
[t]δy(t)dt +
s0i,x [t]δx(t)dµi (t)
t0
i=1
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0.
Introducing the solution formulas into equation (4.5) and combining terms yields
Z tf
0
> 0
l0 ϕxf + σ ψxf Φ(tf )
Φ−1 (t)h1 (t)dt
t
0
Z tf
−1
− l0 ϕ0xf + σ > ψx0 f Φ(tf )
Φ−1 (t)fy0 [t] gy0 [t]
h2 (t)dt
t0
Z t
Z tf
+
l0 fˆ0 [t]Φ(t)
Φ−1 (τ )h1 (τ )dτ dt
t0
t0
Z
Z tf
t
−1
−1
0
0
ˆ
−
l0 f0 [t]Φ(t)
Φ (τ )fy [τ ] gy [τ ]
h2 (τ )dτ dt
t0
t0
(4.6)
Z tf
−1
0
0
−
l0 f0,y [t] gy [t]
h2 (t)dt
t0
Z
Z
ns
tf
t
X
+
s0i,x [t]Φ(t)
Φ−1 (τ )h1 (τ )dτ dµi (t)
−
ns Z
X
i=1
i=1
tf
t0
s0i,x [t]Φ(t)
t0
t0
Z
t
−1
Φ−1 (τ )fy0 [τ ] gy0 [τ ]
h2 (τ )dτ
dµi (t)
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0,
11
LOCAL MINIMUM PRINCIPLE
where
−1 0
0
0
fˆ0 [t] := f0,x
[t] − f0,y
[t] gy0 [t]
gx [t].
Integration by parts yields
Z t
Z tf
Z
Φ−1 (τ )h1 (τ )dτ dt =
fˆ0 [t]Φ(t)
t0
t0
tf
tf
Z
t0
fˆ0 [τ ]Φ(τ )dτ
Φ−1 (t)h1 (t)dt
t
and
tf
Z
fˆ0 [t]Φ(t)
t0
Z
t
Z
−1
Φ−1 (τ )fy0 [τ ] gy0 [τ ]
h2 (τ )dτ
t0
tf Z tf
=
fˆ0 [τ ]Φ(τ )dτ
dt
−1
h2 (t)dt.
Φ−1 (t)fy0 [t] gy0 [t]
t
t0
Introducing these relations into (4.6) leads to
Z tf Z tf
0
> 0
ˆ
l0 ϕxf + σ ψxf Φ(tf ) +
l0 f0 [τ ]Φ(τ )dτ Φ−1 (t)h1 (t)dt
t0
t
Z tf l0 ϕ0xf + σ > ψx0 f Φ(tf )
−
t0
Z tf
−1
ˆ
l0 f0 [τ ]Φ(τ )dτ Φ−1 (t)fy0 [t] gy0 [t]
+
h2 (t)dt
t
Z tf
−1
0
−
l0 f0,y
[t] gy0 [t]
h2 (t)dt
t0
Z
Z
n
s
tf
t
X
+
s0i,x [t]Φ(t)
Φ−1 (τ )h1 (τ )dτ dµi (t)
−
ns Z
X
i=1
t0
i=1
tf
t0
t
Z
s0i,x [t]Φ(t)
Φ
t0
(4.7)
−1
(τ )fy0 [τ ]
−1
gy0 [τ ]
h2 (τ )dτ
dµi (t)
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0,
The Riemann-Stieltjes integrals are to be transformed using integration by parts.
Therefore, let a : [t0 , tf ] → Rn be continuous, b : [t0 , tf ] → Rn absolutely continuous
and µ : [t0 , tf ] → R of bounded variation. Then, using integration by parts for
Riemann-Stieltjes integrals leads to
Z tf
n Z tf
X
a(t)> b(t)dµ(t) =
ai (t)bi (t)dµ(t)
t0
t0
i=1
=
=
n Z
X
tf
Z
bi (t)d
t0
i=1
n
X
t
ai (τ )dµ(τ )
t0
Z
t
tf
ai (τ )dµ(τ ) · bi (t)
t0
i=1
t0
Z
tf
Z
!
ai (τ )dµ(τ ) dbi (t)
t
−
t0
Z
t
=
t0
Z
tf
=
t0
t0
tf Z
−
a(τ )> dµ(τ ) · b(t)
tf
Z
t0
t0
Z
>
a(τ ) dµ(τ ) · b(tf ) −
tf
t0
t
a(τ )> dµ(τ ) b0 (t)dt
t0
Z
t
t0
a(τ ) dµ(τ ) b0 (t)dt.
>
12
M. GERDTS
Application of this formula to (4.7) where a(t)> = s0i,x [t]Φ(t) and
t
Z
t
Z
Φ−1 (τ )h1 (τ )dτ
b(t) =
resp.
b(t) =
t0
−1
Φ−1 (τ )fy0 [τ ] gy0 [τ ]
h2 (τ )dτ
t0
yields
Z
tf
t
Z
s0i,x [t]Φ(t)
Φ
−1
(τ )h1 (τ )dτ
dµi (t)
t0
t0
Z
tf
Z
tf
s0i,x [τ ]Φ(τ )dµi (τ ) Φ−1 (t)h1 (t)dt.
=
t0
t
and
Z
tf
s0i,x [t]Φ(t)
t0
Z
tf
t
Z
t0
tf
Z
=
t0
−1
Φ−1 (τ )fy0 [τ ] gy0 [τ ]
h2 (τ )dτ
dµi (t)
−1
s0i,x [τ ]Φ(τ )dµi (τ ) Φ−1 (t)fy0 [t] gy0 [t]
h2 (t)dt.
t
Substitution into (4.7) yields
Z
tf
t0
l0 ϕ0xf
+σ
>
ψx0 f
Φ(tf ) +
ns Z
X
−
t0
l0 ϕ0xf
+
+σ
>
ns Z
X
i=1
ψx0 f
tf
!
Φ−1 (t)h1 (t)dt
tf
s0i,x [τ ]Φ(τ )dµi (τ )
t
i=1
tf
l0 fˆ0 [τ ]Φ(τ )dτ
t
+
Z
tf
Z
Z
Φ(tf ) +
t
tf
l0 fˆ0 [τ ]Φ(τ )dτ
!
s0i,x [τ ]Φ(τ )dµi (τ )
t
−1
Φ−1 (t)fy0 [t] gy0 [t]
h2 (t)dt
Z tf
−1
0
−
l0 fy,0
[t] gy0 [t]
h2 (t)dt
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0.
(4.8)
Equation (4.8) is equivalent with
Z
tf
>
Z
tf
p1 (t) h1 (t)dt +
t0
p2 (t)> h2 (t)dt
(4.9)
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0,
13
LOCAL MINIMUM PRINCIPLE
where
p1 (t)> :=
Z
l0 ϕ0xf + σ > ψx0 f Φ(tf ) +
tf
l0 fˆ0 [τ ]Φ(τ )dτ
t
+
ns Z
X
tf
!
Φ−1 (t),
s0i,x [τ ]Φ(τ )dµi (τ )
(4.10)
t
i=1
−1
0
p2 (t)> := −l0 fy,0
[t] gy0 [t]
Z
0
> 0
l0 ϕxf + σ ψxf Φ(tf ) +
−
tf
l0 fˆ0 [τ ]Φ(τ )dτ
t
+
ns Z
X
i=1
tf
!
−1
Φ−1 (t)fy0 [t] gy0 [t]
. (4.11)
s0i,x [τ ]Φ(τ )dµi (τ )
t
(4.9) and (3.13) will be exploited in order to derive explicit representations of
the functionals λ∗1 , λ∗2 , and η1∗ . This is possible if either there are no mixed controlstate constraints (1.5) present in the optimal control problem, or if there are no set
constraints (1.7), i.e. U = Rnu .
Corollary 4.1. Let the assumptions of Theorem 3.6 be valid and let there be no
mixed control-state constraints (1.5) in the optimal control problem 1.1. Then there
exist functions p1 ∈ BV ([t0 , tf ], Rnx ) and p2 ∈ L∞ ([t0 , tf ], Rny ) with
λ∗1 (h1 (·))
Z
tf
=−
λ∗2 (h2 (·)) = −
t0
Z tf
p1 (t)> h1 (t)dt,
(4.12)
p2 (t)> h2 (t)dt
(4.13)
t0
for every h1 ∈ L∞ ([t0 , tf ], Rnx ) and every h2 ∈ L∞ ([t0 , tf ], Rny ) . p1 , p2 are given by
(4.10)-(4.11).
Proof. The assertions follow from (4.1), (4.2), (4.9), (4.10)-(4.11) and the preceeding considerations, if we choose h1 (·) = Θ respectively h2 (·) = Θ. p1 is of bounded
variation, since the Riemann-Stieltjes integral in (4.10) is of bounded variation and
Φ, Φ−1 , and the first integral are absolutely continuous. p2 is essentially bounded,
since all terms in (4.11) are so.
Notice, that p1 in (4.10) is even absolutely continuous, if no pure state constraints
(1.6) are present in Problem 1.1.
Corollary 4.2. Let the assumptions of Theorem 3.6 be valid and let U = Rnu .
Let
rank(c0u [t]) = nc
(4.14)
hold almost everywhere in [t0 , tf ]. Furthermore, let the matrix
gy0 [t] − gu0 [t](c0u [t])+ c0y [t]
(4.15)
be non-singular almost everywhere in [t0 , tf ], where
(c0u [t])+ := c0u [t]> c0u [t]c0u [t]>
−1
.
(4.16)
14
M. GERDTS
denotes the pseudo-inverse of c0u [t].
Then there exist functions λ1 ∈ BV ([t0 , tf ], Rnx ), λ2 ∈ L∞ ([t0 , tf ], Rny ), η ∈
∞
L ([t0 , tf ], Rnc ) with
Z tf
∗
λ1 (h1 (·)) = −
λ1 (t)> h1 (t)dt,
λ∗2 (h2 (·)) = −
t0
tf
Z
λ2 (t)> h2 (t)dt,
t0
η1∗ (k(·)) =
Z
tf
η(t)> k(t)dt
t0
∞
for every h1 ∈ L ([t0 , tf ], R ), h2 ∈ L∞ ([t0 , tf ], Rny ), and k ∈ L∞ ([t0 , tf ], Rnc ).
Proof. The assumption U = Rnu implies Uad = L∞ ([t0 , tf ], Rnu ) and hence,
inequality (3.13) turns into the equality
Z tf
0
l0 f0,u
[t]δu(t)dt + η1∗ (c0u [·]δu(·)) − λ∗1 (fu0 [·]δu(·)) − λ∗2 (gu0 [·]δu(·)) = 0
(4.17)
nx
t0
for all δu ∈ L∞ ([t0 , tf ], Rnu ).
Equation (4.9) for the particular choices h1 (·) = fu0 [·]δu(·) and h2 (·) = gu0 [·]δu(·)
yields
Rt
R tf
p (t)> fu0 [t]δu(t)dt + t0f p2 (t)> gu0 [t]δu(t)dt + η1∗ (c0x [·]δx(·) + c0y [·]δy(·))
t0 1
= −λ∗1 (fu0 [·]δu(·)) − λ∗2 (gu0 [·]δu(·)),
(4.18)
where δx and δy are determined by
˙
δx(t)
= fx0 [t]δx(t) + fy0 [t]δy(t) + fu0 [t]δu(t),
0ny = gx0 [t]δx(t) + gy0 [t]δy(t) + gu0 [t]δu(t),
δx(t0 ) = 0nx ,
(4.19)
(4.20)
The latter follows from (4.1), (4.2), and Lemma 3.2.
Introducing (4.18) into (4.17) and exploiting the linearity of the functional η1∗
yields
Z tf
Hu0 [t]δu(t)dt + η1∗ (k(·)) = 0
(4.21)
t0
with
0
Hu0 [t] := l0 f0,u
[t] + p1 (t)> fu0 [t] + p2 (t)> gu0 [t]
and
k(t) := c0x [t]δx(t) + c0y [·]δy(·) + c0u [t]δu(t).
(4.22)
Due to the rank assumption (4.14) equation (4.22) can be solved for δu with
δu(t) = (c0u [t])+ k(t) − c0x [t]δx(t) − c0y [t]δy(t) ,
(4.23)
where (c0u [t])+ denotes the pseudo-inverse of c0u [t]. Using the relation (4.23), equation
(4.21) becomes
Z tf
(4.24)
Hu0 [t](c0u [t])+ k(t) − c0x [t]δx(t) − c0y [t]δy(t) dt + η1∗ (k(·)) = 0.
t0
15
LOCAL MINIMUM PRINCIPLE
Replacing δu in (4.19) and (4.20) by (4.23) yields
˙
δx(t)
= fˆx [t]δx(t) + fˆy [t]δy(t) + fu0 [t](c0u [t])+ k(t),
0ny = ĝx [t]δx(t) + ĝy [t]δy(t) + gu0 [t](c0u [t])+ k(t),
δx(t0 ) = 0nx ,
(4.25)
(4.26)
where
fˆx [t] := fx0 [t] − fu0 [t](c0u [t])+ c0x [t],
fˆy [t] := f 0 [t] − f 0 [t](c0 [t])+ c0 [t],
ĝx [t] :=
ĝy [t] :=
y
u
u
y
0
0
0
+ 0
gx [t] − gu [t](cu [t]) cx [t],
gy0 [t] − gu0 [t](c0u [t])+ c0y [t].
Notice, that ĝy [t] is assumed to be non-singular. Using the solution formula in
Lemma 3.2, the solution of (4.25),(4.26) can be written as
Z t
δx(t) = Φ̂(t)
w(τ )> k(τ )dτ
(4.27)
t0
δy(t) = −ν1 (t)> δx(t) − ν2 (t)> k(t)
(4.28)
where
˙
−1
Φ̂(t) = fˆx [t] − fˆy [t] (ĝy [t]) ĝx [t] Φ̂(t),
Φ̂(t0 ) = Inx
and
−1
w(t)> := Φ̂−1 (t) fu0 [t] − fˆy [t] (ĝy [t]) gu0 [t] (c0u [t])+ ,
−1
ĝx [t],
−1
gu0 [t](c0u [t])+ .
ν1 (t)> := (ĝy [t])
>
ν2 (t) := (ĝy [t])
Equations (4.24), (4.27), and (4.28) and integration by parts yields
Z tf
−η1∗ (k(·)) =
Hu0 [t](c0u [t])+ I + c0y [t]ν2 (t)> k(t)dt
t0
Z
tf
−
Hu0 [t](c0u [t])+ ĉ[t]Φ̂(t)
t0
Z
tf
=
Z
t
>
w(τ ) k(τ )dτ
dt
t0
Hu0 [t](c0u [t])+ I + c0y [t]ν2 (t)> k(t)dt
t0
Z
tf
Z
−
t0
tf
Hu0 [τ ](c0u [τ ])+ ĉ[τ ]Φ̂(τ )dτ
w(t)> k(t)dt
t
where
−1
ĉ[t] := c0x [t] − c0y [t] (ĝy [t])
ĝx [t] = c0x [t] − c0y [t]ν1 (t)> .
With the definition
Z tf
>
0
0
+
η(t) :=
Hu [τ ](cu [τ ]) ĉ[τ ]Φ̂(τ )dτ w(t)> − Hu0 [t](c0u [t])+ I + c0y [t]ν2 (t)>
t
16
M. GERDTS
we thus obtained the representation
η1∗ (k(·))
tf
Z
η(t)> k(t)dt,
=
t0
where η is an element of L∞ ([t0 , tf ], Rnc ).
Introducing this representation into (4.9), setting h2 (·) = Θ, using integration by
parts and the solution formulas (4.3), (4.4) leads to the following representation of
the functional λ∗1 :
Z t
Z tf
Z tf
Φ(τ )−1 h1 (τ )dτ dt
−λ∗1 (h1 (·)) =
p1 (t)> h1 (t)dt +
η(t)> ν3 (t)> Φ(t)
t0
tf
Z
t0
tf
Z
>
=
t0
tf
Z
p1 (t) h1 (t)dt +
t0
Z tf
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ
t0
Φ(t)−1 h1 (t)dt
t
λ1 (t)> h1 (t)dt,
=
t0
where
>
>
Z
tf
λ1 (t) := p1 (t) +
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ
Φ(t)−1
t
and
−1 0
ν3 (t)> := c0x [t] − c0y [t] gy0 [t]
gx [t].
Thus, we obtained the representation
Z
λ∗1 (h1 (·)) = −
tf
λ1 (t)> h1 (t)dt,
t0
where λ1 is an element of BV ([t0 , tf ], Rnx ).
Introducing the representation of η1∗ into (4.9), setting h1 (·) = Θ, using integration
by parts and the solution formulas (4.3), (4.4) leads to the following representation
of the functional λ∗2 :
Z tf −1 ∗
−λ2 (h2 (·)) = −
−p2 (t)> + η(t)> c0y [t] gy0 [t]
h2 (t)dt
t0
tf
Z
−
t0
Z tf
=−
t0
tf
Z
η(t)> ν3 (t)> Φ(t)
−1
Φ(τ )−1 fy0 [τ ] gy0 [τ ]
h2 (τ )dτ
dt
−1 −p2 (t)> + η(t)> c0y [t] gy0 [t]
h2 (t)dt
Z
tf
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ
t0
tf
t0
−
Z
t
Z
−1
Φ(t)−1 fy0 [t] gy0 [t]
h2 (t)dt,
t
λ2 (t)> h2 (t)dt,
=
t0
where
>
>
λ2 (t) := p2 (t) −
η(t)> c0y [t]
−1
gy0 [t]
Z
−
tf
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ
t
−1
Φ(t)−1 fy0 [t] gy0 [t]
.
LOCAL MINIMUM PRINCIPLE
17
Thus, we obtained the representation
λ∗2 (h2 (·)) = −
Z
tf
λ2 (t)> h2 (t)dt,
t0
where λ2 is an element of L∞ ([t0 , tf ], Rny ).
Remark 4.3. The regularity assumptions (4.14) and (4.15) can be replaced by
the assumption that the matrix
0
gy [t] gu0 [t]
c0y [t] c0u [t]
is non-singular a.e. in [t0 , tf ]. Then, Equations (4.20) and (4.22) can be solved w.r.t.
δy and δu and the proof can be adapted.
The remarkable result in the preceeding corollaries is, that they provide useful
representations of the functionals λ∗1 , λ∗2 , and η1∗ . Originally, these functionals are
elements of the dual space of L∞ , which has a very complicated structure. The
exploitation of the fact, that λ∗1 , λ∗2 , and η1∗ are multipliers in an optimization problem,
showed that these functionals actually are more regular.
5. Local Minimum Principles. Theorem 3.6 and Corollary 4.1 yield the following necessary conditions for the optimal control problem 1.1.
The Hamilton function is defined by
>
H(t, x, y, u, λ1 , λ2 , l0 ) := l0 f0 (t, x, y, u) + λ>
1 f (t, x, y, u) + λ2 g(t, x, y, u).
(5.1)
Theorem 5.1 (Local Minimum Principle). Let the following assumptions be
fulfilled for the optimal control problem 1.1.
(i) Let the functions ϕ, f0 , f, g, s, ψ be continuous w.r.t. all arguments and continuously differentiable w.r.t. x, y, and u.
(ii) Let U ⊆ Rnu be a closed and convex set with non-empty interior.
(iii) Let (x̂, ŷ, û) ∈ X be a weak local minimum of the optimal control problem.
(iv) Let Assumption 3.3 be valid.
(v) Let there be no mixed control-state constraints (1.5) in the optimal control
problem 1.1.
Then there exist multipliers l0 ∈ R, σ ∈ Rnψ , λ1 ∈ BV ([t0 , tf ], Rnx ), λ2 ∈ L∞ ([t0 , tf ], Rny ),
and µ ∈ N BV ([t0 , tf ], Rns ) such that the following conditions are satisfied:
(i) l0 ≥ 0, (l0 , σ, λ1 , λ2 , µ) 6= Θ,
(ii) Adjoint equations:
Z
tf
Hx0 (τ, x̂(τ ), ŷ(τ ), û(τ ), λ1 (τ ), λ2 (τ ), l0 )> dτ
t
ns Z tf
X
+
s0i,x (τ, x̂(τ ))> dµi (τ ) in [t0 , tf ],
t
i=1
Hy0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), l0 )> a.e. in [t0 , tf ].
λ1 (t) = λ1 (tf ) +
0ny =
(5.2)
(5.3)
(iii) Transversality conditions:
λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 ,
>
λ1 (tf ) =
l0 ϕ0xf
+σ
>
ψx0 f .
(5.4)
(5.5)
18
M. GERDTS
(iv) Optimality conditions: Almost everywhere in [t0 , tf ] for all u ∈ U it holds
Hu0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), l0 )(u − û(t)) ≥ 0.
(5.6)
(v) Complementarity condition:
µi is monotonically increasing on [t0 , tf ] and constant on every interval (t1 , t2 )
with t1 < t2 and si (t, x̂(t)) < 0 for all t ∈ (t1 , t2 ).
Proof. Under the above assumptions, Corollary 4.1 (with p1 = λ1 , p2 = λ2 )
guarantees the existence of λ1 ∈ BV ([t0 , tf ], Rnx ) and λ2 ∈ L∞ ([t0 , tf ], Rny ) such
that
Z tf
∗
λ1 (h1 (·)) = −
λ1 (t)> h1 (t)dt,
(5.7)
λ∗2 (h2 (·)) = −
t0
tf
Z
λ2 (t)> h2 (t)dt,
(5.8)
t0
holds for every h1 ∈ L∞ ([t0 , tf ], Rnx ) and h2 ∈ L∞ ([t0 , tf ], Rny ).
(a) Equation (3.15) is equivalent with
Z
tf
+
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf )
Z tf
ns Z tf
X
˙
Hx0 [t]δx(t)dt +
s0i,x [t]δx(t)dµi (t) −
λ1 (t)> δx(t)dt
=
t0
t0
i=1
0
t0
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). Application of the computation rules for
Stieltjes integrals yields
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf )
Z tf
Z tf
ns Z tf
X
0
0
+
Hx [t]δx(t)dt +
si,x [t]δx(t)dµi (t) −
λ1 (t)> dδx(t) = 0,
t0
t0
i=1
t0
where
Z
tf
>
λ1 (t) dδx(t) :=
t0
nx Z
X
i=1
tf
λ1,i (t)dδxi (t).
t0
Integration by parts of the last term leads to
l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf )
Z tf
Z tf
ns Z tf
X
+
Hx0 [t]δx(t)dt +
s0i,x [t]δx(t)dµi (t) +
δx(t)> dλ1 (t)
=
0
l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf )
!
Z tf
Z tf
ns Z tf
X
>
0
>
0
>
+
δx(t) d −
Hx [τ ] dτ −
si,x [τ ] dµi (τ ) + λ1 (t)
=
0
t0
i=1
t0
t0
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This is equivalent with
t0
t
i=1
t
19
LOCAL MINIMUM PRINCIPLE
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This implies
λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 ,
>
λ1 (tf ) =
l0 ϕ0xf
+σ
tf
Z
C = λ1 (t) −
>
(5.9)
ψx0 f ,
(5.10)
Hx0 [τ ]> dτ −
t
ns Z tf
X
i=1
s0i,x [τ ]> dµi (τ )
(5.11)
t
for some constant vector C. Evaluation of the last equation at t = tf yields
C = λ1 (tf ), which yields (5.2), (5.4), and (5.5).
(b) Equation (3.12) is equivalent with
Z
tf
Hy0 [t]δy(t)dt = 0
t0
for all δy ∈ L∞ ([t0 , tf ], Rny ). This implies (5.3).
(c) Introducing (5.7),(5.8) into (3.13) leads to the variational inequality
Z
tf
Hu0 [t](u(t) − û(t))dt ≥ 0
t0
for all u ∈ Uad . This implies
Hu0 [t](u − û(t)) ≥ 0
for all u ∈ U for almost all t ∈ [t0 , tf ], cf. Kirsch et al. [13], p. 102. This is
the optimality condition.
(d) According to Theorem 3.6, (3.8), (3.9) it holds η2∗ ∈ K2+ , i.e.
η2∗ (z) =
ns Z
X
i=1
tf
zi (t)dµi (t) ≥ 0
t0
for all z ∈ K2 = {z ∈ C([t0 , tf ], Rns ) | z(t) ≥ 0ns in [t0 , tf ]}. This implies,
that µi is monotonically increasing.
Finally, the condition η2∗ (s(·, x̂(·))) = 0, i.e.
η2∗ (s(·, x̂(·)))
=
ns Z
X
i=1
tf
si (t, x̂(t))dµi (t) = 0,
t0
together with the monotonicity of µi implies that µi is constant in intervals
with si (t, x̂(t)) < 0.
Likewise, Theorem 3.6 and Corollary 4.2 yield the following necessary conditions
for the optimal control problem 1.1. The augmented Hamilton function is defined by
Ĥ(t, x, y, u, λ1 , λ2 , η, l0 ) := H(t, x, y, u, λ1 , λ2 , l0 ) + η > c(t, x, y, u)
(5.12)
Theorem 5.2 (Local Minimum Principle). Let the following assumptions be
fulfilled for the optimal control problem 1.1.
(i) Let the functions ϕ, f0 , f, g, c, s, ψ be continuous w.r.t. all arguments and
continuously differentiable w.r.t. x, y, and u.
20
M. GERDTS
(ii) Let (x̂, ŷ, û) ∈ X be a weak local minimum of the optimal control problem.
(iii) Let
rank (c0u (t, x̂(t), ŷ(t), û(t))) = nc
almost everywhere in [t0 , tf ], cf. Corollary 4.2, (4.14).
(iv) Let the matrix
gy0 [t] − gu0 [t](c0u [t])+ c0y [t]
(5.13)
be non-singular almost everywhere in [t0 , tf ], where
(c0u [t])+ := c0u [t]> c0u [t]c0u [t]>
−1
.
(5.14)
denotes the pseudo-inverse of c0u [t].
(v) Let Assumption 3.3 be valid.
(vi) Let U = Rnu .
Then there exist multipliers l0 ∈ R, σ ∈ Rnψ , λ1 ∈ BV ([t0 , tf ], Rnx ), λ2 ∈ L∞ ([t0 , tf ], Rny ),
η ∈ L∞ ([t0 , tf ], Rnc ), and µ ∈ N BV ([t0 , tf ], Rns ) such that the following conditions
are satisfied:
(i) l0 ≥ 0, (l0 , σ, λ1 , λ2 , η, µ) 6= Θ,
(ii) Adjoint equations:
Z
tf
Ĥx0 (τ, x̂(τ ), ŷ(τ ), û(τ ), λ1 (τ ), λ2 (τ ), η(τ ), l0 )> dτ
t
ns Z tf
X
s0i,x (τ, x̂(τ ))> dµi (τ ) in [t0 , tf ],
(5.15)
+
t
i=1
Ĥy0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), η(t), l0 )> a.e. in [t0 , tf ]. (5.16)
λ1 (t) = λ1 (tf ) +
0ny =
(iii) Transversality conditions:
λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 ,
>
λ1 (tf ) =
l0 ϕ0xf
+σ
>
ψx0 f .
(5.17)
(5.18)
(iv) Optimality conditions: It holds
Ĥu0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), η(t), l0 ) = 0nu
(5.19)
a.e. in [t0 , tf ].
(v) Complementarity conditions:
It holds
η(t)> c(t, x̂(t), ŷ(t), û(t)) = 0,
η(t) ≥ 0nc .
almost everywhere in [t0 , tf ].
µi is monotonically increasing on [t0 , tf ] and constant on every interval (t1 , t2 )
with t1 < t2 and si (t, x̂(t)) < 0 for all t ∈ (t1 , t2 ).
Proof. Corollary 4.2 yields the existence of the functions λ1 , λ2 , η and provides
representations of the functionals λ∗1 , λ∗2 , and η1∗ .
21
LOCAL MINIMUM PRINCIPLE
(a) Equation (3.13) becomes
tf
Z
Ĥu0 [t]δu(t)dt = 0
t0
for all δu ∈ L∞ ([t0 , tf ], Rnu ). This implies (5.19).
(b) Equation (3.15) leads to
Z tf
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) +
Ĥx0 [t]δx(t)dt
t0
Z tf
ns Z tf
X
0
˙
+
si,x [t]δx(t)dµi (t) −
λ1 (t)> δx(t)dt
=
t0
i=1
0
t0
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). Application of the computation rules for
Stieltjes integrals yields
Z tf
0
> 0
0
> 0
l0 ϕx0 + σ ψx0 δx(t0 ) + l0 ϕxf + σ ψxf δx(tf ) +
Ĥx0 [t]δx(t)dt
t0
Z tf
ns Z tf
X
s0i,x [t]δx(t)dµi (t) −
+
λ1 (t)> dδx(t) = 0.
t0
i=1
t0
Integration by parts of the last term leads to
l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf )
Z tf
Z tf
ns Z tf
X
0
0
+
Ĥx [t]δx(t)dt +
si,x [t]δx(t)dµi (t) +
δx(t)> dλ1 (t)
=
0,
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This is equivalent with
l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf )
!
Z tf
Z tf
ns Z tf
X
>
0
>
0
>
+
δx(t) d −
Ĥx [τ ] dτ −
si,x [τ ] dµi (τ ) + λ1 (t)
=
0,
t0
i=1
t0
t
t0
t0
i=1
t
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This implies
λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 ,
λ1 (tf )> =
l0 ϕ0xf + σ > ψx0 f ,
Z tf
ns Z
X
0
>
C = λ1 (t) −
Ĥx [τ ] dτ −
t
i=1
tf
s0i,x [τ ]> dµi (τ )
t
for some constant vector C. Evaluation of the last equation at t = tf yields
C = λ1 (tf ), which proves (5.15).
(c) Equation (3.12) becomes
Z tf
Ĥy0 [t]δy(t)dt = 0
t0
for all δy ∈ L∞ ([t0 , tf ], Rny ). This implies (5.16).
22
M. GERDTS
(d) From
η1∗ ∈ K1+
Z
tf
η(t)> z(t)dt ≥ 0 ∀z ∈ K1
⇔
t0
and
Z
η1∗ (c(·, x̂(·), ŷ(·), û(·))) = 0
tf
⇔
η(t)> c[t]dt = 0
t0
we conclude that
η(t) ≥ 0nc ,
η(t)> c[t] = 0,
a.e. in [t0 , tf ].
According to Theorem 3.6, (3.8), (3.9) it holds η2∗ ∈ K2+ , i.e.
η2∗ (z)
=
ns Z
X
i=1
tf
zi (t)dµi (t) ≥ 0
t0
for all z ∈ K2 = {z ∈ C([t0 , tf ], Rns ) | z(t) ≥ 0ns in [t0 , tf ]}. This implies,
that µi is monotonically increasing.
Finally, the condition η2∗ (s(·, x̂(·))) = 0, i.e.
η2∗ (s(·, x̂(·))) =
ns Z
X
i=1
tf
si (t, x̂(t))dµi (t) = 0,
t0
together with the monotonicity of µi implies that µi is constant in intervals
with si (t, x̂(t)) < 0.
The following considerations apply to both, Theorem 5.1 and Theorem 5.2 and
differ only in the Hamilton functions H and Ĥ, respectively. Hence, we restrict the
discussion to the situation of Theorem 5.1.
The multiplier µ is of bounded variation. Hence, it has at most countably many
jump points and µ can be expressed as
µ = µa + µd + µs ,
where µa is absolutely continuous, µd is a jump function, and µs is singular (continuous, non-constant, µ̇s = 0 a.e.). Hence, the adjoint equation (5.2) can be written
as
Z tf
λ1 (t) = λ1 (tf ) +
Hx0 (τ, x̂(τ ), ŷ(τ ), û(τ ), λ1 (τ ), λ2 (τ ), l0 )> dτ
t
Z tf
ns Z tf
X
s0i,x (τ, x̂(τ ))> dµi,a (τ ) +
s0i,x (τ, x̂(τ ))> dµi,d (τ )
+
t
t
i=1
Z tf
0
>
+
si,x (τ, x̂(τ )) dµi,s (τ )
t
for all t ∈ [t0 , tf ]. Notice, that λ1 is continuous from the right in (t0 , tf ) since µ is
normalized and thus continuous from the right in (t0 , tf ).
23
LOCAL MINIMUM PRINCIPLE
Let {tj }, j ∈ J be the jump points of µ. Then, at every jump point tj ∈ (t0 , tf )
it holds
!
Z tf
Z tf
0
>
0
>
lim
si,x (τ, x̂(τ )) dµi,d (τ ) −
si,x (τ, x̂(τ )) dµi,d (τ )
ε↓0
tj −ε
tj
= −s0i,x (τ, x̂(τ ))> (µi,d (tj ) − µi,d (tj −)) .
Since µa is absolutely continuous and µs is continuous we obtain the jump-condition
λ1 (tj ) − λ1 (tj −) = −
ns
X
s0i,x (tj , x̂(tj ))> (µi (tj ) − µi (tj −)) ,
j ∈ J , tj ∈ (t0 , tf ).
i=1
In order to derive a differential equation for λ1 we need the subsequent auxiliary
result.
Corollary 5.3. Let w : [t0 , tf ] → R be continuous and µ monotonically increasing on [t0 , tf ]. Let µ be differentiable at t ∈ [t0 , tf ]. Then it holds
d
dt
Z
t
w(s)dµ(s) = w(t)µ̇(t).
t0
Proof. Define
Z
t
W (t) =
w(s)dµ(s).
t0
The mean-value theorem for Riemann-Stieltjes integrals yields
Z t+h
W (t + h) − W (t) =
w(s)dµ(s) = w(ξh )(µ(t + h) − µ(t))
t
for some ξh ∈ [t, t + h]. Hence, it follows
W (t + h) − W (t)
w(ξh ) · (µ(t + h) − µ(t))
= lim
= w(t)µ̇(t),
h→0
h→0
h
h
lim
where ξh lies between t and t + h and thus ξh → t for h → 0.
Furthermore, since every function of bounded variation is differentiable almost everywhere, µ and λ1 are differentiable almost everywhere. Putting everything together,
we proved
Corollary 5.4. Let the assumptions of Theorem 5.1 be fulfilled. Then, λ1 is
differentiable almost everywhere in [t0 , tf ] with
λ̇1 (t) = −Hx0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), l0 )> −
ns
X
s0i,x (t, x̂(t))> µ̇i (t).
(5.20)
i=1
Furthermore, the jump conditions
λ1 (tj ) − λ1 (tj −) = −
ns
X
s0i,x (tj , x̂(tj ))> (µi (tj ) − µi (tj −))
i=1
hold at every point tj ∈ (t0 , tf ) of discontinuity of the multiplier µ.
(5.21)
24
M. GERDTS
Notice, that the component µ̇i in (5.20) can be replaced by the absolutely continuous component µ̇i,a , since the derivatives µd and µs vanish almost everywhere.
Similarly, µi in (5.21) can be replaced by the discrete part µd,i , since µa and µs are
continuous. Notice furthermore, that (5.20) and (5.21) provide the absolutely continuous respectively discrete components of λ1 , whereas the singular component of λ1
occurs in (5.2) only.
A special case arises, if no state constraints are present. Then, the adjoint variable λ1 is even absolutely continuous, i.e. λ1 ∈ W 1,∞ ([t0 , tf ], Rnx ), and the adjoint
equations (5.2)-(5.3) become
λ̇1 (t) = −Hx0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), l0 )> a.e. in [t0 , tf ],
0ny = Hy0 (t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), l0 )> a.e. in [t0 , tf ].
(5.22)
(5.23)
The adjoint equations (5.22) and (5.23) form a DAE system of index one for λ1 and
λ2 , where λ1 is the differential variable and λ2 denotes the algebraic variable. This
follows from (5.23), which is given by
>
>
>
0
0ny = l0 f0,y
[t] + fy0 [t] λ1 (t) + gy0 [t] λ2 (t).
Since gy0 [t] is non-singular, we obtain
−> >
>
0
λ2 (t) = − gy0 [t]
l0 f0,y
[t] + fy0 [t] λ1 (t) .
In this section we concentrated on local minimum principles only. The term ‘local’
is due to the fact, that the optimality conditions (5.6) and (5.19), respectively, can
be interpreted as necessary conditions for a local minimum of the Hamilton function
and the augmented Hamilton function, respectively. However, there are also global
minimum principles. In a global minimum principle the optimality condition is given
by
Ĥ(t, x̂(t), ŷ(t), û(t), λ1 (t), λ2 (t), η(t), l0 ) ≤ Ĥ(t, x̂(t), ŷ(t), u, λ1 (t), λ2 (t), η(t), l0 )
for all u ∈ U with c(t, x̂(t), ŷ(t), u) ≤ 0 almost everywhere in [t0 , tf ]. If the Hamilton
function and the function c is convex w.r.t. the control u, then both conditions are
equivalent. Furthermore, the main important difference between local and global
minimum principle is, that U can be an arbitrary subset of Rnu in the global case,
e.g. a discrete set. In our approach we had to assume that U is a convex set with nonempty interior. Proofs for global minimum resp. maximum principles in the context
of ordinary differential equations subject to pure state constraints can be found in,
e.g., Girsanov [6] and Ioffe and Tihomirov [11], pp. 147-159, 241-253.
We favored the statement of necessary conditions in terms of local minimum
principles because these conditions are closer to the finite dimensional case. A connection arises, if the infinite dimensional optimal control problem is discretized and
transformed into a finite dimensional optimization problem. The formulation of the
necessary Fritz-John conditions leads to the discrete minimum principle. These conditions are comparable to the infinite dimensional conditions. This is different for the
global minimum principle. In the finite dimensional case a comparable minimum principle only holds approximately, cf. Mordukhovich [25], or under additional convexity
like assumptions, cf. Ioffe and Tihomirov [11], p. 278.
25
LOCAL MINIMUM PRINCIPLE
6. Regularity. We state conditions which ensure that the multiplier l0 is not
zero and without loss of generality can be normalized to one. Again, we consider
the optimal control problem 1.1 and the optimization problem 3.1, respectively. The
functions ϕ, f0 , f, g, ψ, c, s are assumed to be continuous w.r.t. all arguments and
continuously differentiable w.r.t. x, y, and u. As mentioned before, this implies
that F is Fréchet-differentiable and G and H are continuously Fréchet-differentiable.
According to Corollary 2.2 the following Mangasarian-Fromowitz conditions imply
l0 = 1:
(i) H 0 (x̂, ŷ, û) is surjective.
(ii) There exists some (δx, δy, δu) ∈ int(S − {(x̂, ŷ, û)}) with
H 0 (x̂, ŷ, û)(δx, δy, δu) = ΘZ ,
G(x̂, ŷ, û) + G0 (x̂, ŷ, û)(δx, δy, δu) ∈ int(K).
Herein, the interior of S = W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × Uad is given by
int(S) = {(x, y, u) ∈ X | ∃ε > 0 : Bε (u(t)) ⊆ U a.e. in [t0 , tf ]} ,
where Bε (u) denotes the open ball with radius ε around u. The interior of K1 is given
by
int(K1 ) = {z ∈ L∞ ([t0 , tf ], Rnc ) | ∃ε > 0 : Uε (z) ⊆ K1 }
= {z ∈ L∞ ([t0 , tf ], Rnc ) | ∃ε > 0 : zi (t) ≥ ε a.e. in [t0 , tf ], i = 1, . . . , nc } .
The interior of K2 is given by
int(K2 ) = {z ∈ C([t0 , tf ], Rns ) | z(t) > 0ns in [t0 , tf ]} .
Notice, that the assumption int(K) 6= ∅ in Corollary 2.2 is satisfied for Problem 3.1.
A sufficient condition for (i) to hold is given by
Lemma 6.1. Let Assumption 3.3 be valid and let
rank ψx0 0 Φ(t0 ) + ψx0 f Φ(tf ) = nψ ,
(6.1)
where Φ is the fundamental solution of the homogeneous linear differential equation
Φ̇(t) = A(t)Φ(t),
Φ(t0 ) = Inx ,
t ∈ [t0 , tf ]
and
−1 0
A(t) := fx0 [t] − fy0 [t] gy0 [t]
gx [t].
Then H 0 (x̂, ŷ, û) in Problem 3.1 is surjective.
Proof. Appendix A
Remark 6.2. The rank condition (6.1) together with Lemma 3.2 implies that the
DAE (1.1)-(1.2) is completely output controllable.
Condition (ii) is satisfied if there exist δx ∈ W 1,∞ ([t0 , tf ], Rnx ), δy ∈ L∞ ([t0 , tf ], Rny ),
δu ∈ int(Uad − {û}), and ε > 0 satisfying
c[t] + c0x [t]δx(t) + c0y [t]δy(t) + c0u [t]δu(t) ≤ −ε · e, a.e. in [t0 , tf ],
s[t] + s0x [t]δx(t) < 0ns , in [t0 , tf ],
˙
f 0 [t]δx(t) + f 0 [t]δy(t) + f 0 [t]δu(t) − δx(t)
= 0n , a.e. in [t0 , tf ],
x
y
u
x
gx0 [t]δx(t) + gy0 [t]δy(t) + gu0 [t]δu(t) = 0ny ,
ψx0 0 δx(t0 ) + ψx0 f δx(tf ) = 0nψ ,
a.e. in [t0 , tf ],
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
26
M. GERDTS
where e = (1, . . . , 1)> ∈ Rnc .
Hence, we conclude
Theorem 6.3. Let the assumptions of Theorems 3.6, 5.1 or 5.2, and Lemma 6.1
be fulfilled. Furthermore, let there exist δx ∈ W 1,∞ ([t0 , tf ], Rnx ), δy ∈ L∞ ([t0 , tf ], Rny ),
and δu ∈ int(Uad − {û}) satisfying (6.2)-(6.6).
Then it holds l0 = 1 in Theorems 3.6 and 5.1 or 5.2, respectively.
7. Conclusions and outlook. The presented local minimum principles for optimal control problems subject to index-1 differential-algebraic equations are not only
of theoretical interest but gives rise to numerical solution methods for such problems. The so-called indirect approach intends to fulfill the necessary conditions for
the optimal control problem numerically and thus produces candidates for an optimal
solution. Notice, that the evaluation of the local minimum principle will lead to a
multi-point boundary value problem, at least under suitable simplifying assumptions
such as, e.g., that the structure of active and inactive state constraints is known
and that the singular part of the multiplier µ vanishes, cf. Corollary 5.4. Even for
so-called direct methods, which are based on a discretization of the optimal control
problem, cf. Gerdts [4], the local minimum principle is of great importance in view of
the post-optimal approximation of adjoints.
In the future, it is important to extend these necessary conditions also to higher
index differential-algebraic equations, which do not allow to solve the algebraic equation for y directly. First attempts in this direction show, that it is possible to obtain
likewise results for special classes of index-2 differential-algebraic equations.
Appendix A. Proofs of auxiliary lemmata.
Proof of Lemma 3.2:
Proof. We exploit the non-singularity of B2 (t) and the essential boundedness of
B2 (t)−1 . Equation (3.2) can be solved w.r.t. y:
y(t) = −B2 (t)−1 (A2 (t)x(t) + h2 (t)) .
Introducing this expression into (3.1) yields
ẋ(t) = A1 (t) − B1 (t)B2 (t)−1 A2 (t) x(t) + h1 (t) − B1 (t)B2 (t)−1 h2 (t)
= A(t)x(t) + h(t).
For given x(t0 ) = x0 we are in the situation of Hermes and Lasalle [8], p. 36, and the
assertion follows likewise.
Part (b) exploits the solution formulas in (a). The boundary conditions (3.6) are
satisfied, if
b = C0 x(t0 ) + Cf x(tf )
Z
= C0 x(t0 ) + Cf Φ(tf ) x(t0 ) +
tf
Φ−1 (τ )h(τ )dτ
t0
Z
tf
Φ−1 (τ )h(τ )dτ.
= (C0 + Cf Φ(tf )) x(t0 ) + Cf Φ(tf )
t0
Rearranging terms and exploiting Φ(t0 ) = Inx yields
Z
tf
(C0 Φ(t0 ) + Cf Φ(tf )) x(t0 ) = b − Cf Φ(tf )
t0
Φ−1 (τ )h(τ )dτ.
LOCAL MINIMUM PRINCIPLE
27
This equation is solvable for every b ∈ Rr , if the matrix C0 Φ(t0 ) + Cf Φ(tf ) is of rank
r. Then, for every b ∈ Rr there exists a x(t0 ) satisfying (3.6). Application of part (a)
completes the proof.
Proof of Lemma 3.5 (cf. Ljusternik and Sobolev [18], Lempio [17]):
Proof. Suppose that im(T ) is not closed. Then there exists a sequence (yi , zi ) ∈
im(T ) with limi→∞ (yi , zi ) = (y0 , z0 ) 6∈ im(T ). Then, there exists a (algebraic) hyperplane, which contains im(T ) but not (y0 , z0 ). Hence, there exist a linear functional
ly ∈ Y 0 and a continuous linear functional lz ∈ (Rn )∗ , not both zero, with
(x ∈ X)
ly (T1 (x)) + lz (T2 (x)) = 0,
ly (y0 ) + lz (z0 ) 6= 0.
Notice, that ly is not necessarily continuous. In fact, we will show that ly is actually
continuous. Therefore, let ηi ∈ Y be an arbitrary sequence converging to zero in Y .
Unfortunately, there may be many points in the pre-image of ηi under the mapping
T1 . To circumvent this problem, instead we consider the factor space X/ker(T1 ) whose
elements [x] are defined by [x] := x+ker(T1 ) for x ∈ X. The space X/ker(T1 ) endowed
with the norm k[x]kX/ker(T1 ) := inf{ky − xkX | y ∈ ker(T1 )} is a Banach space. Notice
that ker(T1 ) is a closed subspace of X.
If we consider T1 as a mapping from the factor space X/ker(T1 ) onto Y , then
the inverse operator of this mapping is continuous, because T1 is surjective and X
and Y are Banach spaces. Hence, to each ηi ∈ Y there corresponds an element Wi
of X/ker(T1 ). Since the inverse operator is continuous and ηi is a null-sequence, the
sequence Wi converges to zero. Furthermore, we can choose a representative ξi ∈ Wi
such that
kξi kX ≤ 2kWi kX/ker(T1 ) ,
T1 (ξi ) = ηi
hold for every i ∈ N. Since Wi is a null-sequence the same holds for ξi by the first
inequality.
This yields
lim ly (ηi ) = lim ly (T1 (ξi )) = lim −lz (T2 (ξi )) = 0,
i→∞
i→∞
i→∞
since ξi → ΘX and T2 and lz are continuous. This shows, that ly is actually continuous
in ΘX and hence on X.
In particular, we obtain
0 = lim (ly (yi ) + lz (zi )) = ly
i→∞
lim yi + lz lim zi = ly (y0 ) + lz (z0 )
i→∞
i→∞
in contradiction to 0 6= ly (y0 ) + lz (z0 ).
Proof of Lemma 6.1:
Proof. Let h1 ∈ L∞ ([t0 , tf ], Rnx ), h2 ∈ L∞ ([t0 , tf ], Rny ), and h3 ∈ Rnψ be given.
Consider the boundary value problem
H10 (x̂, ŷ, û)(δx, δy, δu)(t) = h1 (t),
H20 (x̂, ŷ, û)(δx, δy, δu)(t) = h2 (t),
H30 (x̂, ŷ, û)(δx, δy, δu) = h3 .
a.e. in [t0 , tf ],
a.e. in [t0 , tf ],
28
M. GERDTS
In its components the boundary value problem becomes
˙
−δx(t)
+ fx0 [t]δx(t) + fy0 [t]δy(t) + fu0 [t]δu(t) = h1 (t),
gx0 [t]δx(t) + gy0 [t]δy(t) + gu0 [t]δu(t) = h2 (t),
0
ψx0 δx(t0 ) + ψy0 0 δy(t0 ) + ψx0 f δx(tf ) + ψy0 f δy(tf ) = −h3 .
a.e. in [t0 , tf ],
a.e. in [t0 , tf ],
By the rank assumption and Assumption 3.3 all assumptions of
Lemma 3.2 are satisfied and the boundary value problem is solvable. This shows
the surjectivity of the mapping H 0 (x̂, ŷ, û).
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