Optimization of Nonlinear Singularly Perturbed Systems with Hypersphere Control Restriction

1. BSU Optimization of Nonlinear Singularly Perturbed Systems with Hypersphere Control Restriction 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 the values of which are bounded in the Euclidean metrics: 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. 2. Statement of the Problem In the class of multidimensional controls with piecewise-continuous components we consider the time-optimal problem where is a small positive parameter, is a -dimensional vector, is a -dimensional vector, the other elements of the problem have the appropriate dimensions, and the following assumptions are made: Assumption 1. Matrix 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 with piecewise-continuous components and the corresponding trajectory of system (1) are said to be asymptotically suboptimal (subextremal) if and the following asymptotic equalities hold: where 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. (1) (2)

3. 3. Algorithm The calculations begin by solving the reduced problem where 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 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 According to the Pontryagin maximum principle where Assumption 3. The vector of conjugate variables which corresponds to the extremal is uniquely determined up to a positive multiplier, and , When this assumption holds, as can be seen from formula (3), (3) (4)

4. The next stage of the algorithm is to solve the optimal control problem with process of infinite duration which henceforth will be called the second basic problem. The specific feature of this problem is as follows: the point is the equilibrium position of the dynamical system for the control 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 and is normal. In accordance with the maximum principle where and 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 (5) (6) (7) (8)

5. 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 is an asymptotically subextremal control. Note that it can be formed immediately after solving the basic problems. (9) 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.