Distributed Optimal Control Of A Mesoscopic Model And The New Mechanochemical Model

The development of optimal control theory originated from great demands for modelling of real life applications in economics,engineering and applied mathematics in the early 1950's.Optimal control theory of partial differential equations had been used widely in science,engineering and biology,such as,temperature control[65]and the control of Ginzburg-Landau model used in the study of superconductivity[48]and so on.In the thesis,we consider distributed optimal control problem of a simplified meso-scopic model and the new mechanochemical model as well as the existence of weak solution to a simplified mesoscopic model with degenerate mobility.In the first chapter,we investigate the initial-boundary value problem of a simpli-fied mesoscopic model with degenerate mobility in n space dimensionsUsing a Galerkin method,we obtain the existence of weak solution to the regulariza-tion problem corresponding to problem(1).By taking the limits of weak solution to the regularization problem,we prove the existence of weak solutions to problem(1).Observing that mobility D(u)is under a more general assumptions,when we take the limits of weak solution to the regularization problem,we encounter difficulties in taking the limits of the terms with mobility.Using the method of constructing the set,we can take the limits of the terms with mobility in the set.Different from the literature[29],we improve the regularity of weak solution and obtain the relationship of l and u in some set.In the second chapter,we study the distributed optimal control problem for a simplified mesoscopic model with state constraints subject to state constraints and the state system where k is a positive constant,f(y)=y3-y and Bu is the control term.Using a Galerkin method,we prove the existence and uniqueness of strong solution to the state system(4)-(6)as well as the continuity of dependence on the control variable.Since the cost functional(2)we consider is discontinuous,together with state constraint(3),these makes it difficult to directly establish the necessary optimality conditions.The difficulty we have met is how to choose some suitable spaces for optimal control problem.Using for reference on the definition of new cost functional in[86,101],we study the optimal control problem(2)-(6)by constructing a new penalty functional to approximate the cost functional(2).Here we use a different method in[101]to derive the necessary optimality con-ditions for the approximating optimal control problem.Finally by considering the limits of the necessary optimality conditions we have obtained,we derive the necessary optimality conditions of the problem(2)-(6).Different from the literature[101],we improve the space of definition of the functionals g(t,y).In the third chapter,we study optimal control problem for the mechanochemical model in biological patterns with state constraint subject to state constraints and the state system where ?>0 and B1? and B2? are the control terms.We extend the method of the second chapter to study the optimal control of systems of higher-order,since the system of equations has a higher order equation,it makes our research much more complicated than the second chapter,especially when we derive the necessary optimality conditions for the approximating optimal control problem.As we know,there exists a few results on the study of optimal control problem for systems of higher-order with state con-straints.Hence,the main difficulty we meet is how to construct some suitable spaces for optimal control problem as well as when the space variable of the functionals g and h are extended to two,we verify whether some properties of the functionals g and h hold.Different from the literature[96],we improve the space of definition of the functionals g(t,u,v).In the fourth chapter,we study the distributed optimal control for the new mechanochemical model.That is,the control problem consists in minimizing the cost functional,denoted by(CP)subject to the control constraint and the state system where ?>0,?1,?2,?3 are given nonnegative constants but not all zero,?Q? L2(Q),???L2(?)are given target functions,uumin ? L?(Q)and umax ? L?(Q)are given functions satisfying umin?Umax,a.e.in Q.Different from the model[18],the model we consider has no energy functional.We also have met the difficulty of obtaining a prior estimates,which is caused by the term ?3.Using a Galerkin method,we prove the existence of weak solution to the state system(15)-(18)and prove the existence and uniqueness of strong solution to the state system(15)-(18)as well as the continuity of dependence on the control variable.By obtaining the existence and uniqueness of solution to the linearized system as well as some stability estimates of solution,we prove the differentiability of the control-to-state operator S.Finally we derive the first-order necessary optimality conditions for(13)-(18).