Random Coefficient And Linear Stochastic Differential System With Jump H < Sub > 2 < / Sub > / H < Sub > Up < / Sub > Control

The thesis is concerned with H∞control and H2/H∞control of linear stochastic differential systems with random coefficients or with jumps.Chapter1reviews the theories of H∞control and H2/H∞control, and linear quadratic optimal control. Using a Nash equilibrium approach, the existence of so-lutions to H2/H∞control is converted into that of solutions to the Riccati equation, while the Riccati equation associated with H∞robustness is indefinite.Chapter2is concerned with the H2/H∞control of linear stochastic differential systems with random coefficients and only state-dependent noise. By using Bell-man’s quasilinear principle and a method of monotone convergence, we prove the existence and uniqueness of solutions to indefinite backward stochastic differential equation(BSRE), thus obtaining the bounded real lemma with random coefficients. Consequently, we present sufficient and necessary conditions for the existence of H2/H∞control in terms of a pair of coupled BSREs.Chapter3is concentrated on a bounded real lemma of linear stochastic dif-ferential systems with random coefficients and state-and control-dependent noise. We prove that there exists some constant such that when the norm of the input-output operator is less than it, the associated LQ problem is solvable. We then use the solution of stochastic Hamiltonian system to construct that of the indefinite BSRE. As a special case of the bounded real lemma with random coefficients, we give the bounded real lemma with Markov jumping parameters. We also consider the existence of solution to a class of special BSRE arising from H∞robustness.Chapter4is firstly concerned with a bounded real lemma for linear stochastic differential systems driven by a Brownian motion and a Poisson point process with deterministic coefficients. Using the existence of local solutions of ordinary differ-ential equation and the essential technique of quasilinear principle, we obtain the global solution of the indefinite Riccati equation. We point out that in the Ito system driven by a Brownian motion and a Poisson point process with random coefficients, we can prove the bounded real lemma when the martingale part is independent of control. We then prove the equivalence between the existence of H2/H∞control and that of the solutions to four coupled matrix-valued equations. In the end, when the system is driven by a Brownian motion and Poisson point process with Markovian jumping parameters, we prove the associated bounded real lemma.