Optimization of Nonlinear Singularly Perturbed Systems with
Hypersphere Control Restriction
BSU
A.I. Kalinin and J.O. Grudo
Belarusian State University, Minsk, Belarus
1. Introduction
Within the framework of the theory of optimal control, great attention is given to the singularly
perturbed problems. As is known, the numerical solution of optimal control problems entails repeated
integration of the original and conjugate systems. In singularly perturbed problems, these dynamical
systems are stiff, and, as a rule, the computations are associated with serious difficulties resulting in
large computation time and accumulation of computation errors. Therefore, the role of asymptotic
methods is growing, especially in view of the fact that the use of these methods results in
decomposition of the original optimal control problem into problems of lower dimension.
In the report, we consider a time-optimal problem for a nonlinear singularly perturbed systems with
multidimensional control
u  t   u1  t  , , ur  t  the values of which are bounded in the
Euclidean metrics:

u  t   u12  t  

 ur2  t   C.
An algorithm for the construction of asymptotic approximation to the solution to the problem in
question is proposed. The algorithm employs solutions to two optimal control problems of lower
dimension than the original problem.
2. Statement of the Problem

In the class of multidimensional controls u  t   u1  t  ,
components we consider the time-optimal problem
y  a1  y, t   A1  y, t  z  B1  y, t  u,
, ur  t   with piecewise-continuous
y  0   y0 ,
 z  a2  y, t   A2  y, t  z  B2  y, t  u, z  0   z0 ,
y T   0,
z T   0,
u  t   1, t   0, T  , J  u   T  min,
(1)
(2)
where  is a small positive parameter, y is a n -dimensional vector, z is a m -dimensional vector, the
other elements of the problem have the appropriate dimensions, and the following assumptions are
made:
Assumption 1. Matrix A2  y, t  , y  Rn , t  0, is Hurwitz, that is, the real parts of all its
eigenvalues are negative.
Assumption 2. All functions forming the problem are twice continuously differentiable.
Before proceeding, we need to first define the asymptotic approximations to the solution to
problem (1), (2).
Definition. A control u  t ,   , t  0, T     , with piecewise-continuous components and the
corresponding trajectory x  t ,    y  t ,   , z  t ,   , t  0, T     , of system (1) are said to


be asymptotically suboptimal (subextremal) if u  t ,    1, t  0, T     , and the following


asymptotic equalities hold:


xT  ,    O1  , T    T 0    O2  
where T 0   is the optimal time (the final time of a Pontryagin extremal) in problem (1), (2).
In this report, we describe an algorithm by means of which an asymptotically subextremal control
can be constructed for the problem in question.
3. Algorithm
The calculations begin by solving the reduced problem
y  f 0  y, u, t  , y  0   y0 , y T   0,
u  t   1, t  0, T  , J 0  u   T  min,
where
f0  y, u, t   a0  y, t   B0  y, t  u, a0  a1  A1 A21a2 , B0  B1  A1 A21B2 .
Henceforth this will be called the first basic problem.
The purpose of using numerical methods to solve nonlinear problems is not so much to find an
optimal control as to find a Pontryagin extremum. We therefore assume that the extremal
u 0  t  , t  0, T0  , has been constructed as a result of solving the first basic problem. The
corresponding trajectories of the direct and conjugate systems will be denoted by
y0  t  ,  0  t  , t 0, T0 .
According to the Pontryagin maximum principle
0  t  u 0  t   max 0  t  u, t   0, T0  ,
u 1

(3)

where 0  t    0  t  B0 y 0  t  , t , t  0, T0 .
Assumption 3. The vector of conjugate variables  0  t  , twhich
to the extremal
 0, Tcorresponds
0 ,
0
u  t  , tis uniquely
0, T0  , determined up to a positive multiplier, and
0  t   0 ,
t  0, T0 .
When this assumption holds, as can be seen from formula (3),
u 0  t    0  t   0  t  , t   0, T0 .
(4)
The next stage of the algorithm is to solve the optimal control problem with process of infinite
duration
dz ds  A2  0, T0  z  B2  0, T0  u , z  0   A21  0, T0  a2  0, T0  ,
z      A21  0, T0  B2  0, T0   0 T0 
J u  
0
    T  u  s  
0
0

 0 T0  , u  s   1, s  0,
(5)

 0 T0  ds  max .
which henceforth will be called the second basic problem. The specific feature of this problem is as
follows: the point  A21  0, T0  B2  0, T0   0 T0   0 T0  is the equilibrium position of the
dynamical system for the control
(6)
u  s    0 T0   0 T0  ,
which makes the integrand in the quality criterion vanish. In particular, this implies that the second
basic problem has a solution if an optimal control exists in a similar problem with a finite sufficiently
long process.
Assumption 4. Problem (6) has a solution u*  s  , s  0, and is normal.
In accordance with the maximum principle
(7)
  s    0 T0   u *  s   max   s    0 T0   u, s  0,


where   s      s  B  0, T  and
2
0
d ds   A2  0, T0  .
u 1


  s  , s  0, is a solution of the conjugate system
If this assumption is satisfied, as follows from formula (7), the optimal control in the second basic
problem has the form
u *  s      s    0 T0     s    0 T0  , s  0.
(8)
We can prove, using the boundary functions method, that under Assumptions 1 – 5 a Pontryagin
extremal exists in the original problem with sufficiently small  and the vector function
u0  t ,   
0  t     s 
0  t     s 
, s
t  T0

, t  0, T0  ,
(9)
is an asymptotically subextremal control. Note that it can be formed immediately after solving the
basic problems.
4. Conclusions
The proposed algorithm asymptotically solves the problem under consideration. It is essential that
its realization presupposes the decomposition of the original problem into two problems of lower
dimension, and what is more, the algorithm does not contain integrations of stiff systems.