Convergence of Galerkin approximations for operator Riccati equations - a nonlinear evolution equation approach. (English) Zbl 0717.65064
This paper is concerned with the convergence of Galerkin approximations to infinite-dimensional operator Riccati differential equations. The author considers a class of generalized operator Riccati equations in the space of Hilbert-Schmidt operators on a separable Hilbert space. Here a formulation of the problem due to R. Temam [J. Funct. Anal. 7, 85- 115 (1971; Zbl 0218.47032)] is employed. In this context, assuming that the underlying Hilbert space admits a densely, continuously and compactly embedded subspace and the sequence of approximating Galerkin subspaces have orthogonal projections strongly convergent, the author proves norm convergence of the Galerkin approximating Riccati operators, uniformly in time on compact time intervals.
Finally the theory is applied to a linear quadratic optimal control problem for a one-dimensional heat equation.
Finally the theory is applied to a linear quadratic optimal control problem for a one-dimensional heat equation.
Reviewer: M.Calvo
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65J10 | Numerical solutions to equations with linear operators |
| 65K10 | Numerical optimization and variational techniques |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 34G10 | Linear differential equations in abstract spaces |
Keywords:
convergence; Galerkin approximations; infinite-dimensional operator Riccati differential equations; Hilbert-Schmidt operators; Hilbert space; linear quadratic optimal control problem; heat equationCitations:
Zbl 0218.47032References:
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