EXACT, APPROXIMATE AND S-CONTROLLABILITY OF SEMILINEAR CONTROL SYSTEMS

Abstract

The thesis concerns exact and approximate controllability of abstract semilinear deterministic control systems and S-controllability of non-deterministic semilinear control systems. Let V and V be Hilbert Spaces and Z = L2 [to, T:17] and Y = L2 [to , T:r2] be the corresponding function spaces defined on [t0 , T], 0 t0 < t T < cc. Consider the semilinear control system dxu (t) Axu(t)+Bu(t)+ f(t,x„(t)) dt (1) xu (to )= xo where A:D(A)cV —* V is a closed linear operator which generates a Co -semigroup S(t), B:Vs ---->V is a bounded linear operator and f:[to,T]xV-->V is a nonlinear operator. xu (t) is the state value at time t E [to , T] corresponding to the control u taken from the control space Y. In Chapter 3, the exact controllability of abstract semilinear control system (1) has been proved under the conditions that the linear operator A is negative definite and the nonlinear operator f satisfies the Lipschitz condition. As particular cases, the exact controllability has been proved for controlled heat equation, wave equation, system of infinite ordinary differential equations and mechanical system.