| This dissertation investigates mainly some problems of feedback controls for some integral/differential equations with singular properties.We discuss the linear quadratic optimal control problems for some finite dimensional and infinite dimensional singular Volterra integral equations,and obtain the causal state feedback representations for the open-loop optimal control,respectively.We are also concerned with the application of the linear quadratic optimal control theory for some ordinary differential equations in the study of blowup controllability with feedback controls.The blowup at finite time is also a singular property of the equations.This dissertation consists of three parts.In Chapter 1,we introduce the background of the problems of feedback controls and present the main conclusions of this dissertation.In Chapter 2,we are concerned with the linear quadratic optimal control problems for some finite dimensional and infinite dimensional singular Volterra integral equations.Under some necessary convexity conditions,an optimal control exists,and can be characterized via Fréchet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions.However,these(equivalent)characterizations are not causal,meaning that the current value of the optimal control depends on the future values of the optimal state.Practically,this is not feasible.In Chapter 2,the cases that the cost functional to be just convex(plus a proper range condition)and uniformly convex are considered,respectively.We obtain the causal state feedback representation of the optimal control via a Fredholm integral equation,by which we mean that the current value of the optimal control can be written in terms of the current values of the optimal state and the past values of the optimal state.Moreover,we see that fractional differential equations are special cases of the singular Volterra integral equations.Then,our framework can cover LQ problems for fractional differential equations.In Chapter 3,we are concerned with the application of the linear quadratic optimal control theory for some ordinary differential equations in the study of blowup controllability with feedback controls.We first obtain the feedback approximate null controllability and feedback null controllability by using the classical results of the linear quadratic optimal control problems for ordinary differential equations.We can obtain the results of the approximate null controllability by bounded feedback operators.However,one can not make the ordinary differential system be feedback null controllable by a bounded feedback operator due to the backward solvability of ordinary differential equations.Thus,we obtain the results of the null controllability by unbounded feedback operators.Then,the global exact blowup controllability and approximate blowup controllability with feedback controls are derived on the time-invariant ordinary differential system,respectively.The results of approximate blowup controllability with feedback controls can guarantee the feedback operator to be bounded at any time before blowup.Finally,we concern the robustness of blowup controllability with feedback controls on the time-invariant system.The research on the blowup controllability with feedback controls for ordinary differential equations in this paper also lays a theoretical foundation for further considering the blowup controllability with feedback controls for parabolic equations in the future. |
