Constructive Methods of Optimal
Control under Uncertainty
Rafail Gabasov
Belarussian State University
NATO ARW, October 21-25
Outline
Introduction
1. Classical optimal feedback and its realization
2. Optimal guaranteeing feedbacks
3. Optimal control under imperfect information
4. Optimal decentralized control
5. Parallelizing of computations during optimal control of large systems
6. Optimal on-line control with delays
7. Optimal control of time-delay systems
8. Optimal control of PDEs
9. Nonlinear optimal control problems
10. Stabilization of dynamical systems
Gabasov
2
NATO ARW, October 21-25
Introduction
Points of view on Optimal Control Theory
 Calculus of Variations
 Control Theory
Principles of control
 Open-loop control
 Closed-loop control
 Real time (on-line) control
Types of closed loops
 Feedforward
 Feedback
 Feedforward-feedback (combined)
Gabasov
3
NATO ARW, October 21-25
Linear optimal control problem
cx(t * )  max
x  A(t ) x  B (t )u , x(t* )  x0
(1)
x(t * )  X * , u (t )  U , t  T  [t* , t * ]
x  x(t )  R n : state of the control system at the instant t
u  u (t )  R r : value of control at the instant t
A(t )  R nn , B(t )  R nr , t T : piecewise continuous matrix functions
X *  {x  R n : g*  Hx  g *} : terminal set (g*, g*Rm, HRmn)
U  {u  R r : u*  u  u * } : set of accessible values of control
Discrete controls (h=(t* – t*)/N, N>0):
u (t )  u ( ), t  [ ,  h[,   Th  {t* , t*  h,t *  h}
Gabasov
4
(2)
NATO ARW, October 21-25
Classical optimal feedback
Imbed problem (1) into a family
cx(t * )  max
x  A(t ) x  B(t )u , x( )  z
(3)
x(t * )  X * , u (t ) U , t  T ( )  [ , t * ]
depending on Th and zRn
u0(t|, z), tT() : optimal open-loop control of (3) for a position (, z)
X : set of states z for which optimal open-loop control exists
Optimal feedback :
u 0 ( , z )  u 0 ( |  , z ), z  X  ,   Th
Gabasov
5
(4)
NATO ARW, October 21-25
Realization of optimal feedback
Real system closed by optimal feedback:
x  At x  Bt u 0 (t , x)  w, xt*   x0
(5)
u 0 (t , x)  u 0 (t , x(t ))  u 0 ( , x( )), t  [ ,  h[,   Th
w : disturbances
Trajectory of (5) is a solution to linear differential equation:
x  At x  Bt u (t )  w, xt*   x0 , u (t )  u 0 (t , x(t ))
(6)
Particular control process with w*(t), tT:
x * (t )  At x* (t )  Bt  u 0 (t , x* (t ))  w* (t ), x* t*   x0 , t  T (7)
Realization of optimal feedback in a particular control process:
u * (t )  u 0 ( , x* ( ))  u 0 ( |  , x* ( )), t  [ ,  h[,   Th
Gabasov
6
(8)
NATO ARW, October 21-25
Optimal Controller
:
cx(t * )  max, x  A(t ) x  B (t )u , x( )  x* ( )
(9)
x(t * )  X * , u (t ) U , t  T ( )
Linear programming problem:
 c' (t )u (t )  max, g* ( ) 
tTh ( )
*
D
(
t
)
u
(
t
)

g
( )

(10)
tTh ( )
u*  u (t )  u * , t  Th ( )  Th  [ , t * ]
t h
c' (t ) 
 c ( ) B( )d 
t
t h
D(t ) 
 ( ) B( )d 
t
t h
*
1
c
'
F
(
t
)
F
( ) B( )d ,

t
t h
*
1
HF
(
t
)
F
( ) B( )d , t  Th ;

t
*
g* ( )  g*  ( ) x* ( ), g ( )  g *  ( ) x* ( )
Gabasov
 c '   c ' A(t ),  (t * )  c;
7
  A(t ), (t * )  H

F  A(t ) F , F (t* )  E
NATO ARW, October 21-25
Fast algorithms for optimal open-loop control

Gabasov R., Kirillova F.M. (2001)
Fast algorithms for positional optimization of dynamic
systems. Proceedings of the Workshop "Fast solutions of
discretized optimization problems". (K.-H.Hoffmann, R.Hoppe
and V. Schulz eds.)

Gabasov R., Kirillova F.M. and N.V. Balashevich (2000).
On the Synthesis Problem for Optimal Control Systems. SIAM
J. Control Optim.

Gabasov, R., F.M. Kirillova and N.V. Balashevich (2000).
Open-loop and Closed-loop Optimization of Linear Control
Systems. Asian Journal of Control.
Gabasov
8
NATO ARW, October 21-25
Analysis
 – h : u0(t| – h, x*( – h)), tT( – h)
cx(t * )  max
0
x
( )
*
x  A(t ) x  B (t )u , x(  h)  x (  h)
x* ( )
x* (  h)
x(t * )  X * , u (t )  U , t  T (  h)
x 0 (t * )
u 0 ( |   h,...)
u*(t) = u0( – h | – h, x*( – h)), t[ – h,[ : control fed into the system
w*(t), t[ – h,[ : realized disturbance
 : u0(t| , x*( )), tT( )
cx(t * )  max
x  A(t ) x  B (t )u , x( )  x * ( )
x(t * )  X * , u (t )  U , t  T ( )
Gabasov
9
NATO ARW, October 21-25
Example :
optimal damping of two-mass oscillating system
25
 u (t )dt  min
0
x1  x3 , x 2  x4 , x3   x1  x2  u , x 4  0.1x1  1.01x2
x1 (0)  x2 (0)  0, x3 (0)  2, x4 (0)  4
x1 (25)  x2 (25)  x3 (25)  x4 (25)  0
0  u (t )  1, t  [0,25]
Real system :
x1  x3 , x2  x4 , x3   x1  x2  u, x4  0.1x1  1.01x2  w
with disturbance :
w * (t )  0.3sin 4t , t [0,9.75[; w * (t )  0, t  9.75;
Gabasov
10
NATO ARW, October 21-25
Example :
optimal damping of two-mass oscillating system
Gabasov
11
NATO ARW, October 21-25
Discussion

Direct control system




No state or mixed constraints





Gabasov R., Dmitruk N.M. and F.M.Kirillova (2004).
Indirect Optimal Control of Dynamical Systems. Comput. Math. Math. Phys.
Gabasov R., Kirillova F.M. and N.S. Pavlenok (2003).
Design of Optimal Feedbacks in the Class of Inertial Controls.
Automation and Remote Control
Gabasov R., Kirillova F.M. and N.N.Kovalenok (2004)
Synthesis of optimal signals for the control of dynamical systems with
Lipschitz bang-bang actuators. Dokl. Akad. Nauk, Ross. Akad. Nauk
Gabasov R., F.M. Kirillova and N.V. Balashevich (2001).
Algorithms for open-loop and closed-loop optimization of control
systems with intermediate state constraints. Comput. Math. Math. Phys.
Information on disturbances is not used
Exact measurements of all states are available
Mathematical model with lumped parameters, not large
Problem is linear
Gabasov
12
NATO ARW, October 21-25
Optimal guaranteeing feedbacks
cx(t * )  max
x  A(t ) x  B (t )u  d (t ) w, x(t* )  x0
(11)
x(t * )  X * , u (t ) U , t  T  [t* , t * ]
w  w(t ), t T : disturbance
W  {w  R : w*  w  w*} : set of possible values of the disturbance
Types of feedback :


Gabasov
unclosable

closable

closed
R.Gabasov, F.M.Kirillova and N.V.Balashevich (2004).
Guaranteed on-line control for linear systems under disturbances.
Functional Differential Equations
13
NATO ARW, October 21-25
Example :
optimal guaranteeing feedbacks
x2 (t * )  max
x1  x2 , x 2   x1  u  w
x(0)  x0* , x1 (t * )  X *  {x1  R : x1*  x1  x1*}
u (t )  1, w(t ) W  {w  R : | w | w*}, t  T  [0, t * ]
Parameters : x0*  (0,1), t *  12, x1*  2, x1*  7, w*  0.5
Guaranteed values of the
performance index:
1) unclosable feedback
2) one-time closable feedback
with closure instant t = 8
2
1
Gabasov
14
NATO ARW, October 21-25
Control System under Disturbances
T=[t*,t*] : control interval
Dynamical system with disturbance :
x  At x  Bt u  M t w
(12)
M t   R nnw : piecewise continuous matrix function
Measuring device :
y  C t x  
(13)
C t   R qn : continuous matrix function
y  yt   R q : output
   t   R q : errors of the measuring device
Measurements are made at discrete instants


t  Th  t* , t*  h,t *  h
Gabasov
15
NATO ARW, October 21-25
Elements of uncertainty
Initial state :
x(t* )  X 0 : X 0  x0  GZ
(14)
x0  R n , G  R nnz : given
Z  z  R nz : d*  z  d * : set of possible values of parameters z


Disturbance :
wt    l t vl  t v, t  T
(15)
lL
l t , l  L : piecewise continuous functions (L={1,2,…nv})
v  R nv : vector of parameters of the disturbance

V  v  R nv : d*  v  d *

: set of possible values of parameters v
Measurement errors :
*   (t )   * , t  Th ;
Gabasov
 *  *  0
16
(16)
NATO ARW, October 21-25
Classical control of the system under uncertainty
t = t* :



measurement y(t*) is obtained (generated by x(t*), ξ(t*))
vector u(t*) = u(t*,y(t*)) U is chosen
control function u(t) = u(t*), t [t*,t*+h[, is fed into the system
t = t*+h :
system moves to the state x(t*+h)
 measurement y(t*+h) is obtained (generated by x(t*+h), ξ(t*+h))
…….

t=:

measurement y( ) is obtained
signal y ()  ( y(t ), t  Th ( )  {t* , t*  h,, }) is formed

vector u( ) = u(, y(·)) U is chosen


control function u(t) = u( ), t [ , +h[, is fed into the system
Y (u ) : totality of all signals y(·) that can be obtained under chosen u
Y (u )  Yt* (u )
Gabasov
17
NATO ARW, October 21-25
Optimal classical feedback
Feedback under inaccurate measurements :
u  u , y , y   Y (u ),   Th
(17)
X (t , u, y), t T : set of all trajectories of
x  At x  Bt ut , yt   M t w
for a chosen feedback u and a fixed signal y(·)=(y(t), tTh)
X (t , u)   X (t , u, y), y() Y (u)
(18)
Admissible feedback u  u , y , y   Y (u ),   Th :
X (t * , u )  X *
Performance index : J u   min cx, x  X (t * , u )
Optimal (guaranteeing) feedback :
u 0  u 0  , y , y   Y (u 0 ),   Th :
Gabasov
18
 
J u 0  max J (u )
NATO ARW, October 21-25
Optimal on-line control
Suppose that by the moment  :
*
*
*
 measurements y t* , y t*  h , , y   h  has been made
*
*
*
 controls u t* , u t*  h , , u   h  has been calculated in time
st* , st*  h , , s  h  (neglected for simplicity)
 control function
u * (t )  u *  , t  [ ,  h[,   Th (  h)
has been fed into the system
At the moment  :
 current measurement y*( ) is obtained
Aim :
*
0
*
 calculate current value of control u ( )  u ( , y ())
 feed to the input of control object the control function
u * (t )  u *  , t  [ ,  h[
Gabasov
19
NATO ARW, October 21-25
A priori and a posteriori distribution sets
A priori distribution sets :
Z : a priori distribution set of parameters z of the initial state x(t*)
V : a priori distribution set of parameters v of the disturbance w(t), tT
=ZV=(γ=(z,v) : z  Z, v V) : a priori distribution
of unknown parameters γ of the system
A posteriori distribution set :
ˆ    ˆ  ; y* ()


set of all vectors γ to which there correspond the initial condition
x(t*)=x0+Gz and the disturbance w(t)=Λ(t)v, t[t*,[, able together with
some measurement error ξ(t), tTh(), to generate the signal y* ()
Gabasov
20
NATO ARW, October 21-25
Admissible open-loop control (program)
*
Function u (t ), t  [ , t ], is said to be an admissible open-loop control if
*
together with u (t ), t  [t* , [, it transfers the control system (12) at the
moment t* on the terminal set X* for all γ from ̂ 
Equivalent to :
The admissible control u (t ), t  [ , t * ], transfers the determined system
x  A(t ) x  B(t )u, x( )  x0* ( )
(19)
x0* ( ) : state of this system with x(t*)=x0, u(t)=u*(t), t[t*,[
at the moment t* to the terminal set:
X * ( )  {x  R n : g* ( )  Hx  g * ( )}
(20)
g* ( )  ( g*i ( )  g*i  *i  , i  I ), g * ( )  ( g i* ( )  g i*   i*  , i  I ); I  {1,..., m}
Gabasov
21
NATO ARW, October 21-25
Accompanying optimal observation problems
*
To establish admissibility of control u (t ), t  [ , t ], it is required to solve
extremal problems
 *i    min  pxi z  pwi v , ( z , v)  ˆ ( )
    max  pxi z  pwi v , ( z , v)  ˆ ( )
*
i
(21)
pxi  hi  F (t * ), pwi  hi  P(t * )
hi  : i-th row of matrix H
F  At F , F t*   G; P  At P  M t t , Pt*   0
Problems (21) are called optimal observation problems accompanying
the optimal control problem under uncertainty (accompanying optimal
observation problems)
Gabasov
22
NATO ARW, October 21-25
Optimal open-loop control
and accompanying optimal control problem
The quality of the admissible open-loop control is evaluated by
I u   min cx(t * ),   ˆ ( )
Optimal open-loop control u 0 (t |  ,  ( )), t [ , t * ] solves problem
cx(t * )  max
(22)
x  A(t ) x  B(t )u, x( )  x0* ( )
x(t * )  X * ( ), u (t ) U , t  T  [ , t * ]
called optimal control problem accompanying the optimal control
problem under uncertainty (accompanying optimal control problem)
Let
Gabasov
u * ( )  u 0 ( , y* ())  u 0 ( |  ,  ( ))
23
NATO ARW, October 21-25
Scheme of optimal on-line control of
dynamical system under uncertainty
At the moment  :
 Solve 2m accompanying optimal observation problems
 Solve the accompanying optimal control problem
OE : Optimal Estimator
solves accompanying
optimal observation problem
OC : Optimal Controller
solves accompanying
optimal control problem
Gabasov
24
NATO ARW, October 21-25
Optimal observation problems

Gabasov R., Dmitruk N.M., Kirillova F.M. (2002).
Optimal Observation of Nonstationary Dynamical Systems.
Journal of Computer and Systems Sciences Int.

Gabasov R., Dmitruk N.M., Kirillova F.M. (2004).
Optimal Control of Multidimensional Systems by Inaccurate
Measurements of Their Output Signals. Proceedings of the
Steklov Institute of Mathematics.
Gabasov
25
NATO ARW, October 21-25
Example :
optimal control under imperfect information
Mathematical model :
mx  k1  k 2 x  k1l1  k 2l2   u1  u 2  w1
m  k1l1  k 2l2 x  k1l12  k 2l22   l1u1  l2u2  w2


Control interval : T=[0,15]
Parameters : m  1, l1  1.1, l2  0.9, k1  1.1, , k 2  1.1, J  1 / 3
Initial condition :
x0  0.1,  (0)  0, x 0  z1 ,  (0)  z 2
 z1 , z 2   Z  z  R 2 : z1  0.1, z 2  0.33
Disturbance : w1 t   v1 sin 4t , w2 t   v2 sin 3t , t  T
v1 , v2   V  v  R 2 : vi  0.01, i  1,2
Sensor : y1   x  l1  1 , y 2  x  l2   2
 i t   0.01, t  Th  0, h,,15  h, h  0.02
Gabasov
26
NATO ARW, October 21-25
Example :
optimal control under imperfect information
Performance index :
15
J u    u1 t   u 2 t dt  min
0
Terminal condition :
 x15, x 15  X *  x  R 2 : x1  0.05, x2  0.1
 15,  15   *    R 2 : 1  0.05,  2  0.2
Particular process :
z1*  0.1, z 2*  0.33, v1*  0.005, v1*  0.01
1* t   0.01cos 2t ,  2* t   0.01cos 4t , t  Th
Gabasov
27
NATO ARW, October 21-25
Example :
optimal control under imperfect information
Gabasov
28
NATO ARW, October 21-25
Example :
optimal control under imperfect information
Gabasov
29
NATO ARW, October 21-25
Optimal decentralized control
Optimal control of a group of q objects:
 cixi (t * )  max
iI
xi  Ai (t ) xi 
 Aij (t ) x j Bi (t )ui   Bij (t )u j ,
jI i
 H i xi (t * )  g ,
iI
jI i
xi (t* )  xi 0 , i  I
ui (t ) U i , t  T
xi  xi (t )  R ni , ui  ui (t )  R ri , i  I  {1,2,, q},  ni  n,  ri  r
iI
Aij (t )  R
ni n j
, Bij (t )  R
ni r j
iI
, Ai (t )  Aii (t ), Bi (t )  Bii (t ), i, j  I , t  T
H i  R mni , i  I , g  R m ; U i  {ui  R ri : ui*  ui  ui*}
Gabasov
30
NATO ARW, October 21-25
Optimal decentralized control
For control of q subsystems q Optimal Controllers operating in parallel are used:
u1*
u q*
DS1

DS q
OC q

OC1
xq*
x1*
At each moment  Th
i-th Optimal Controller obtains:

xi* ( ) : current state of i-th subsystem

results, obtained by all other OCs at
previous moment  – h
Realization of optimal feedback :
ui* ( )  ui0 ( |  , xi* ( ), x*j (  h), i  I i ),   Th , i  I
ui0 (t |  , xi* ( ), x*j (  h), i  I i ), t  T ( ) : optimal open-loop control
of problem with ri inputs
Gabasov
31
NATO ARW, October 21-25
Example : optimal decentralized control
12
 (u1(t )  u2 (t ))dt  min
0
x1  x2 , x2  1.98 x1  x3  u1, x1 (0)  0.5, x2 (0)  0
x3  x4 , x4  0.25 x1  1.23 x3  0.5u2 , x3 (0)  0.2, x4 (0)  0
x1 (12)  x2 (12)  x3 (12)  x4 (12)  0
0  u1 (t )  0.5, 0  u2 (t )  0.2, t  [0,25]
x1  x2 , x2  1.98x1  x3  u1  w1, x3  x4 , x4  0.25x1  1.23x3  0.5u2  w2
w1* (t )  0.6 sin 4t , w2* (t )  0.05 sin 3t , t  [0,6], w1* (t )  w2* (t )  0, t ]6,12]
1
2
2
1
1) decentralized
2) centralized
Gabasov
32
NATO ARW, October 21-25
Parallelizing of computations during optimal control of
large systems
cx(t * )  max
(1)
x  A(t ) x  B (t )u , x(t* )  x0
x(t * )  X * , u (t )  U , t  T  [t* , t * ]
e.g., two systems :
l
xi   aik (t ) xk ai (t ) xl 1  bi (t )u , xi (t* )  xi 0 , i  1, l
k 1
xi 
Gabasov
n
 aik (t ) xk ai (t ) x1  bi (t )u,
k l 1
33
xi (t* )  xi 0 , i  l  1, n
NATO ARW, October 21-25
Quasidecomposition of the fundamental matrix




F (t )  





f11 (t )
f 21 (t )

f l1 (t )
f l 11 (t )
f l  21 (t )

f n1 (t )








f1n (t )   p11
f 2 n (t )    21
 
   
f l n (t )   l1

 pl 11

f l 1n (t )
f l 1n (t )  l  21
   
f nn (t )    n1








p1n
2n

l n
pl 1n
l  2 n

 nn




  F~ (t )





l
(23)
ij   aik (t ) kj ai1 (t ) p1 j (t )  ai (t ) pl 1 j (t ), ij (t* )   ij , i  2, l
(24) ij 
Gabasov
k 2
 ii  1, i  1, n;  ij  0, i, j  1, n, i  j
n
 aik (t ) kj ai (t ) p1 j (t )  ail 1 (t ) pl 1 j (t ), ij (t* )   ij ,
i  l  2, n
k l  2
34
NATO ARW, October 21-25
Example :
parallelizing of computations
25
 u (t )dt  min, x1  x3 ,
x 2  x4 , x3   x1  x2  u , x 4  0.1x1  1.01x2
0
x1 (0)  x2 (0)  0, x3 (0)  2, x4 (0)  4, x1 (25)  x2 (25)  x3 (25)  x4 (25)  0
0  u (t )  1, t  [0,25]
Number of parameters
of approximation of F(t)
Terminal state
Value of performance index
40
0.00102613
0.00410210
-0.00028247
-0.00129766
7.041025176
50
-0.00017364
0.00015934
-0.00010114
0.000031591
7.047378611
No approximation (exact)
(10-7, 10-7, 10-7, 10-7)
7.046875336
Gabasov
35
NATO ARW, October 21-25
Optimal on-line control with delays
Every Optimal Controller calculates u*() in time  = Kh, K>1

OC1
u 0 (t | t* , x* (t* )), t  [t* , t * ]
OC 2
u 0 (t | t*  h, x* (t*  h)), t  [t*  h, t * ]


OC q
u 0 (t |  , x* ( )), t  [ , t * ]
t* t*  h
t*   t*    h


u 0 (t ),
t  [t* , t*    h[
 u 0 (t | t , x* (t )),
t  [t*    h, t*    2h[
*
*
*
u (t )   0
*
u
(
t
|
t

h
,
x
(t*  h)), t  [t*    2h, t*    3h[
*




Gabasov
36
NATO ARW, October 21-25
Optimal control of time-delay systems
cx(t * )  max, x (t )  A(t ) x(t )  e1a (t ) x1 (t   )  B (t )u
x(t* )  x0 , x1 (t )  x10 (t ), t  [t*   , t*[
(25)
x(t * )  X * , u (t ) U , t  T  [t* , t * ]
a(t)R, tT; x10(t)R, t[t*–,t*[ : piecewise continuous functions
 : delay; e=(1,0,…,0) Rn
Optimal feedback :
u 0 ( , z ())  u 0 ( |  , z ()), z ()  Z ,   Th
z ()  ( z  R n , z1 (t )  R, t  [   , [) : state of system
u0(t|, z(·)), tT( ) : optimal open-loop control of (25) for (, z(·))
Realization of optimal feedback :
u * ( )  u 0 ( , x* ()), x* ()  ( x* ( ), x1* (t ), t  [   , [),   Th
Gabasov
37
NATO ARW, October 21-25
Quasireduction of the fundamental matrix
 f11 (t )
F (t )   
 f (t )
 n1
f12 (t ) 


f n 2 (t ) 
f1n (t ) 
 
f nn (t ) 

 p11 (t ) 12 (t )  1n (t ) 
~
 


   F (t )



(
t
)


(
t
)
p
(
t
)
n2
nn
 n1

pi1(t), tT : finite-parametric approximations of fi1(t), tT , i  1, n
φij(t), tT : solutions to n – 1 systems of ODE’s
n
ij    ik akj (t )  pi1 (t )a1 j (t ), ij (t * )   ij , i  1, n,
k 2
j  2, n
R.Gabasov, O.Yarmosh. Fast algorithm of open-loop solution in a linear optimal
control problem for dynamical systems with delays. Today, Section C-1.
Gabasov
38
NATO ARW, October 21-25
Example:
optimal control of time-delay systems
5
 u(t )dt  min
0
x1 (t )   x1 (t  1)  u, x1 (0)  x10 , x1 ( s )  x10 ( s ), s  [1;0[
x2 (t )  x1 (t ), x2 (0)  x20 , | x1 (5) | 0.1, | x2 (5) | 0.1; 0  u (t )  0.5, t  T  [0,5]
Real system :
x1(t )   x1(t  1)  u  w, x2 (t )  x1(t )
,
*
with disturbances : a) w (t )  0.2 sgn sin( 2t ), t  T
b) w* (t )  0.2, t . T ; c) w* (t )  0.2, t  T
Gabasov
39
NATO ARW, October 21-25
Optimal control of PDEs
Problem of optimal heating
t*
(26)
 | u (t )   | dt  min
0
x( s, t )
 2 x ( s, t )
*
a
,
(
s
,
t
)

Q

{
0

s

l
,
0

t

t
}
2
t
s
x( s, t )
x( s, t )
 0,
  (u (t )  x(l , t )), x( s,0)  x0 ( s ), s  S  [0, l ]
s s 0
s s l
l
g*   h( s ) x( s, t * )ds  g * , u*  u (t )  u * , t  T
0
x(s,t), (s,t)Q : temperature at point s at instant t
u(t), tT : control
a, ν , α, β>0 : given constants; h(s)Rm, sS; g*, g*Rm
Gabasov
40
NATO ARW, October 21-25
Approximation of PDE
Sη={0, η,…, l- η, l}, l/K, K>0
ys(t)=x(s,t), sSη, tT
t*
 | u (t )   | dt  min
0
(27)
y  Ay  bu , y (0)  y0
g*  Hy (t * )  g * , u (t )  1, t  T
y0=(x0(s), sSη), H =(h(s), sSη), b=(0,0,…,0,aν/η)
0
 1 1
 1 2 1
a 0
1 2
A 2
   
0
0
0
0
0
0

Gabasov
 0
0 
 0
0 
 0
0 
 
 
 2
1 
 1    1
41
NATO ARW, October 21-25
Nonlinear optimal control problems

Nonlinear dynamics
cx(t * )  max
x  f ( x)  bu , x(0)  x0
x(t * )  X * , | u (t ) | 1, t  T  [0, t * ]
f(x), xX :

Gabasov
N.V. Balashevich, R. Gabasov, A.I. Kalinin, and F.M. Kirillova (2002).
Optimal Control of Nonlinear Systems. Comp. Mathematics and
Math.Physics
42
NATO ARW, October 21-25
Nonlinear optimal control problems

Nonlinear performance index
t*
 ( x(t * ))   f 0 ( x(t ))dt  min, x  Ax  bu , x(0)  x0
0
x(t * )  X * , | u (t ) | 1, t  T  [0, t * ]
φ(x), f0(x), xX : convex functions

Nonlinear input
cx(t )  max
cx(t * )  max
x  f ( x)  b( x)u , x(0)  x0
x  f ( x)  b(u ), x(0)  x0
x(t )  X , | u (t ) | 1, t  T
x(t * )  X * , | u (t ) | 1, t  T
*
*

*
Arbitrary set U, convex terminal set X*
Gabasov
43
NATO ARW, October 21-25
Stabilization of dynamical systems

Gabasov R. Kirillova F.M. and O.I. Kostyukova (1995).
Dynamic system stabilization methods. Journal of Computer
and Systems Sciences International.

Gabasov, R.F.; Ruzhitskaya, E.A. (1999).
A method of stabilization of dynamic systems under persistent
perturbations. Cybernetics and Systems Analysis
Gabasov
44
NATO ARW, October 21-25