Controllability And Time Optimal Controls For Multiplicative Control Systems Governed By Heat Equations

In this paper,we study the controllability and the time optimal control problems for the multiplicative control systems governed by the linear and semilinear heat equations with the homogenous Neumann boundary condition.Moreover,the bang-bang prop-erty of time optimal controls for the linear heat equation with bilinear controls and the homogenous Dirichlet boundary condition are obtained.In the first part of the paper,the(exact)controllability of arbitrary constant aim is proved for the multiplicative control system of the linear heat equation with the homoge-nous Neumann boundary condition.As we all know,the L2-norm of the solution for the heat equation with the homogenous Neumann boundary condition decays to the steady-state(which is the mean of the initial state,and is a constant)with time exponentially[cf.SIAM J.Appl.Math.1978]).Combining the controllability of the traditionally ad-ditive locally distributed controls for the heat equation with the homogenous boundary condition and the cost estimates of the control functions,we reach the constant aim by designing a proper step-by-step control.These results essentially rely on the homoge-nous boundary condition and coincide with the real of the heat-transfer system with the domain being perfectly insulated,and they differ significantly from those of the homoge-nous Dirichlet boundary condition case.Moreover,we also prove the negative results of the controllability based on the maximal principle of the parabolic equations.It should be pointed that there are many works on the null controllability on the mul-tiplicative control system governed by the heat equation with the homogenous Dirichlet boundary condition[SIAM J.Control Optim.41(2003)1886-1900,Z.Angew.Math.Mech.(1)87(2007),14-23].As we all know,the L2-norm of the solution for the heat equation with the homogenous Dirichlet boundary condition decays with time exponen-tially.Thus,the solution y(·,t)of the multiplicative control system of the heat equation with the homogenous Dirichlet boundary condition can arrive at any neighborhood of zero-state if we take the control function to be zero and the time to be large enough.So,we can restart from a state near zero to reach zero-state by choosing a proper control function.For the heat equation with the homogenous Neumann boundary condition,the solution will converge to the steady-state ye.However,the aim function is not neces-sarily equal to ye.Therefore,we need to overcome more difficulties in order to prove the controllability of the multiplicative control system with the homogenous Neumann boundary condition.Although there is no limit on the control function which we find by using the step-by-step method,it may take too long a time to reach the target in our first result.This is a remarkable difference compared with the parabolic system with traditionally addi-tive locally distributed controls,in which the aim state can be achieved of the moment.Therefore,the existence of the time optimal control is getting increasingly urgent for multiplicative system.We will investigate the existence of the time optimal control for multiplicative system in Chapter ?,and the L?-norm of the control function is limited to be no more than a positive constant.The system we considered is governed by semilin-ear parabolic equations.Since no growth restriction is imposed on the nonlinearity f(s),in general,it has no globally defined(in time)solution,and the solution may blow up in finite time.Therefore,we need to ensure the existence time of the local solution in time by analyzing the relation among the existence time of the solution,the L?-norm of the control function and the nonlinearity of f(s)carefully.Then,based on these estima-tions,we prove the existence of the time optimal control by some standard compactness criteria.