Optimal boundary control and new Riccati equations for highly damped second-order equations
Roberto Triggiani · Differential and Integral Equations · 1994
We study the quadratic cost, optimal control problem over an infinite time horizon of a new abstract system, driven by an analytic semigroup.The model is motivated by, and encompasses in particular, a whole class of wave-like and plate-like partial differential equations defined on a bounded domain with high internal damping (which causes a parabolic-like behavior), and with control on the boundary in L2 (0, oo; L2 (r) ).The paper contains a thorough study of the corresponding algebraic Riccati equation, and of the pointwise feedback synthesis of the optimal pair.The new model introduces additional conceptual and technical difficulties over the boundary control case for more traditional parabolic dynamics of the literature [2], [13], [14], and these are reflected by the novelties in the final results as well.1. Introduction; parameterized mathematical model; statement of main results.Though this paper is written at the abstract level, its motivation comes from, and is ultimately directed to, applications to non-standard partial differential equations of parabolic type, with control function acting on the boundary of the spatial domain.See Section 4, where in addition appropriate references to the literature will be made.Accordingly, in this paper we consider the following mathematical model.1.1.Abstract parameterized system.Let Y (state) and U (control) be two Hilbert spaces.The dynamics which we consider is given by the following input-solution relationship: