Linear Quadratic Regulatory Boundary/Point Control of Stochastic Partial Differential equation Systems with Unbounded Coefficients

In this dissertation, an infinite dimensional stochastic linear quadratic control problem is studied in both finite and infinite horizons in an abstract setting. The goal is to extend the results obtained by DaPrato and Flandoli to the systems with singular estimates, which is in the center of the abstract framework developed and extensively studied by Lasiecka, Triggiani, Avalos and Bucci. The abstract setting consists of the unbounded control operator B, the bounded (multiplicative) noise operator C and the dynamics generator A, which generates a strongly continuous semigroup. The operators A, B, and C are all deterministic and acting on spaces of Hilbert space valued random variables. This problem has been studied in the literature within the context of analytic semigroups e At and the corresponding stochastic processes driven by relatively (w.r.t. A) bounded control operators B. This condition, in which A generates an analytic semigroup, is relaxed and replaced by the singular estimate condition for the pair (A,B). This covers some significant coupled PDE systems subject to point/boundary control which arise in applications containing interactions of hyperbolic and parabolic dynamics. The associated stochastic Riccati equation has unbounded coefficients as a result of infinite dimensional dynamics and the relatively-bounded control operator. We provide examples both simple and examples fully representing our approach. The main results are full optimal feedback syntheses along with wellposedness of stochastic Riccati equation for an infinite dimensional stochastic linear quadratic control problem with unbounded control operators for both the finite and the inifinite horizon cases. In the infinite horizon case, as a by-product the stabilizing optimal feedback is obtained.

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