Linear-quadratic optimal control of hereditary differential systems: Infinite dimensional Riccati equations and numerical approximations. (English) Zbl 0557.49017
The author discusses the linear quadratic optimal control problem for retarded functional differential systems in the Hilbert state space \(M^ 2\) on finite and infinite time intervals. After a review of the general theory in Hilbert spaces, he shows that solutions of the integral and algebraic Riccati equations are of trace class. Then strong and weak convergence of approximate Riccati equations to solutions of the infinite dimensional Riccati equation are discussed. These results are applied to delay equations and the so-called averaging approximation is treated in detail. For approximation on the infinite time interval a property is needed (conjecture 7.1) which ensures uniform exponential stability of the approximating semigroups provided that the system semigroup is uniformly exponentially stable. In the meantime, this conjecture has been confirmed by D. Salamon [Structure and stability for approximations of functional differential equations, to appear in: SIAM J. Control Optimization]. Finally, computational aspects are discussed and three numerical examples are presented.
Reviewer: F.Colonius
MSC:
49M15 | Newton-type methods |
34K35 | Control problems for functional-differential equations |
65J10 | Numerical solutions to equations with linear operators |
93B40 | Computational methods in systems theory (MSC2010) |
49K99 | Optimality conditions |
93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
93D15 | Stabilization of systems by feedback |
15A24 | Matrix equations and identities |