Averaging method in optimal control problems for integro-differential equations

2.1 Optimal control problem, nonlinear with respect to the control, for a system of integro-differential equations with rapidly oscillating parameters

We consider the nonlinear control problem for a system of integro-differential equations with rapidly oscillating parameters:

with the quality criterion

over the interval [ 0 , T ] , where ε > 0 is a small parameter, T > 0 is a given constant, x is the state vector in R d , u ( t ) is the m -dimensional control vector such that u ( t ) ∈ W ⊂ R m , d , m = 1 , 2 , 3 , … , Φ ( x ) is a given function.

The function x ε ( t , u ) denotes the solution of the Cauchy problems (1) and (2), corresponding to the control u ( t ) . For simplicity of notation, in the following discussion, we omit the explicit dependence on u and ε and denote this solution as x ( t ) .

We assume that there exists a function X 0 ( x , u ) such that for all x ∈ R d and u ∈ W , the following limit exists uniformly:

where

The optimal control problems (1) and (2) with rapidly oscillating coefficients correspond to a simpler optimal control problem

with the quality criterion

For problems (1) and (2), we assume that the following conditions hold:

1. The admissible controls are considered to be m -dimensional vector functions u ( ⋅ ) such that u ( ⋅ ) ∈ U , where U is a compact set in L 2 ( 0 , T ) ;

2. The function X ( t , x , y , u ) is defined and jointly continuous in all its variables in the domain Q 0 = { t ≥ 0 , x ∈ R d , y ∈ R n , u ∈ W } , and satisfies:

   1. a linear growth condition with respect to x , y in Q 0 ; that is, there exists a constant M > 0 such that

      ∣ X ( t , x , y , u ) ∣ ≤ M ( 1 + ∣ x ∣ + ∣ y ∣ ) ,

      for any ( t , x , y , u ) ∈ Q 0 ;

   2. a Lipschitz condition with constant λ in Q 0 ; that is,

      ∣ X ( t , x , y , u ) − X ( t , x 1 , y 1 , u 1 ) ∣ ≤ λ ( ∣ x − x 1 ∣ + ∣ y − y 1 ∣ + ∣ u − u 1 ∣ ) ,

      for all ( t , x , y , u ) , ( t , x 1 , y 1 , u 1 ) ∈ Q 0 ;

3. The function φ ( t , s , x ) is defined and continuous in the domain Q 1 = { t ∈ [ 0 , T ] , s ∈ [ 0 , T ] , x ∈ R d } , takes on the values in R n , and satisfies the linear growth condition and the Lipschitz condition with respect to x , i.e., there exists L φ > 0 such that

   ∣ φ ( t , s , x ) ∣ ≤ L φ ( 1 + ∣ x ∣ ) and ∣ φ ( t , s , x ) − φ ( t , s , x 1 ) ∣ ≤ L φ ∣ x − x 1 ∣ ;

4. There exists the limit (3) uniformly in x ∈ R d and u ∈ W ;

5. The function L ( t , x , u ) is defined in the domain Q 2 = { t ∈ [ 0 , T ] , x ∈ R d , u ∈ W } , and

   1. L ( t , x , u ) is uniformly continuous in x ∈ R d with respect to t ∈ [ 0 , T ] and u ∈ W ;

   2. L ( t , x , u ) satisfies the Lipschitz condition with respect to u in Q 2 , with constant λ > 0 ;

   3. The function Φ : R d → R is continuous in x .

The conditions (C2), (C3), and Theorems 3.1 [12] and 2.2 [13] imply that for any admissible control u ( t ) , there exists a unique solution x ( t , u ) of the Cauchy problem on the whole interval [ 0 , T ] . It is hence obvious that problems (1) and (4) are valid for all admissible controls.

The main result of this subsection is the theorem that establishes the relationship between the optimal control and the quality criteria of the exact problems (1), (2) and the averaged problems (4), (5). We set

2.2 Optimal control problem, linear with respect to the control, for a system of integro-differential equations with rapidly oscillating parameters

We also consider the control problem with rapidly oscillating parameters, that is, linear with respect to the control input:

with the quality criterion

over the interval [ 0 , T ] , where ε > 0 is a small parameter, T > 0 is a given constant, x ∈ R d is the state vector, and u ( t ) is the m -dimensional control vector belonging to a functional set.

If the following limit exists uniformly with respect to x ∈ R d :

with

then the optimal control problems (8), (9) with rapidly oscillating coefficients correspond to a simpler control problem on the interval [ 0 , T ] :

with the corresponding quality criterion

The main result is the proof of the convergence of the minimal values of the quality criterion, optimal controls, and optimal trajectories of the exact problems (8) and (9) to the corresponding minimal values of the quality criterion, optimal controls, and trajectories of the averaged problems.

We assume that the following conditions are met for problems (8) and (9):

1. The admissible control is an m -dimensional vector function u ( ⋅ ) ∈ L p ( ( 0 , T ) ; V ) , p > 1 , taking on the values in a closed convex set V ⊂ R m ;

2. The function f ( t , x , y ) is defined and jointly continuous in all its variables in the domain Q 3 = { t ≥ 0 , x ∈ R d , y ∈ R n } ; the n × m matrix function f 1 ( x ) is defined for x ∈ R d , and

   1. f ( t , x , y ) satisfies the linear growth condition with constant M in the domain Q 3 , i.e., ∣ f ( t , x , y ) ∣ ≤ M ( 1 + ∣ x ∣ + ∣ y ∣ ) for all ( t , x , y ) ∈ Q 3 ;

   2. f ( t , x , y ) and f 1 ( x ) satisfy, with respect to x , the Lipschitz condition with constant λ > 0 in their domains;

3. Function φ ( t , s , x ) is defined and continuous in the domain Q 4 = { t ∈ [ 0 , T ] , s ∈ [ 0 , T ] , x ∈ R d } , takes on the values in the space R n , and satisfies, with respect to x , the linear growth condition and the Lipschitz condition; that is, there exists some L φ > 0 such that

   ∣ φ ( t , s , x ) − φ ( t , s , x 1 ) ∣ ≤ L φ ∣ x − x 1 ∣ and ∣ φ ( t , s , x ) ∣ ≤ L φ ( 1 + ∣ x ∣ ) ;

4. Limit (10) exists uniformly in x ∈ R d ;

5. The scalar functions A ( t , x ) and B ( t , u ) are defined for t ∈ [ 0 , T ] , x ∈ R d , u ∈ V , and jointly continuous in all their variables, and

   1. A ( t , x ) ≥ 0 , B ( t , u ) ≥ a ∣ u ∣ p with a constant a > 0 , for all t ∈ [ 0 , T ] , and the function B ( t , u ) is convex with respect to u ∈ V ;

   2. The function Φ : R d → R is non-negative and continuous in x .

The main result here is the following theorem on a relationship between the optimal triples of the exact and averaged problems.