Numerical Algorithms For Distributed Elliptic Optimal Control Problems

The distributed optimal control problems are the infinite-dimensional control systems governed by partial differential equations or partial differential-integral equations or coupling of partial differential equations and ordinary differential equations. These problems have been widely applied in many engineering technology such as space-flight technology, civil engineering, ecosystem and community system. The research of numerical algorithms for distributed optimal control problems is an important branch in the optimal control problems. At present, the numerical algorithms for distributed optimal control problems involve principally the finite difference technique, multi-grid method, finite element approximation and mixed finite element method. In this thesis, the numerical algorithms for distributed elliptic optimal control problems are mainly studied, which involve mixed finite element method, boundary element method, coupled method of finite element and boundary element and optimal equilibrium solution for solving the multi-objective control problems. The theoretic proof, algorithm organization and error analysis of these numerical algorithms are obtained and several test examples are presented to illustrate the results, respectively, in which we present some new algorithms or improve the best algorithms presently known. Main contributions are as following.1. Mixed finite element methodA numerical iterative method based on mixed finite element method for optimal boundary control problem governed by bi-harmonic equation is presented. We introduce the costate equation with clamped boundary conditions, and develop the system of optimality equations consisting of state and costate function. Prom the optimality system we discover that the vortex function relative to the costate equation is just equal to some multiple of the control function on the boundary in the optimal sense, which automatically generate a suitable gradient. Based on the suitable gradient, a gradient-type optimization method using a mixed finite approximation is developed. In our algorithm, the trace of vortex function of costate equation on boundary plays a key role. Furthermore a local error analysis at every iterative for this method is given. We apply this algorithm to the solution of a test problem.2. Boundary element methodAn optimality system of equations for the optimal control problem governed by Helmholz-type equations is derived under some convexity conditions. By the associated first-order necessary optimality condition, we obtain the conjugate gradient method (CGM) in the continuous case. Introducing the sequence of higher-order fundamental solutions of Helmholtz equation in two dimensions, we propose an iterative algorithm based on the conjugate gradient-boundary element method using the multiple reciprocity method (CGM+MRBEM) for solving the discrete control input. This algorithm has an advantage over that of the existing literatures because the main attribute (the reduced dimensionality) of the boundary element method is fully utilized. Furthermore the local error estimates for this scheme are obtained, and a test problem is given to illustrate the efficiency of the proposed method.Introducing the sequence of higher-order fundamental solutions of fourth-order plate equations on Winkle foundation in two dimensions, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the multiple reciprocity-boundary element method (MRBEM) is developed for the optimal control problem associated with the state equation governed by fourth-order elliptic boundary value problem. The local error estimates based on the stability of this scheme in the H~2 norm, L~2 norm and L~∞norm are obtained and a numerical example is given.3. Coupled method of finite element and boundary elementThe coupling of finite element and boundary element solution for the optimal control of the stationary solid fuel ignition model associated with the state equation governed the boundary value problem with the exponential nonlinearity term is investigated. The associated first-order necessary optimality condition is derived. Introducing the fundamental solution of Laplace equation, we develop the boundary integral equations for the optimality system. We look upon the exponential nonlinearity term as a unitary functions to constitute the finite element approximations for the domain functions, which can derive the system of nonlinear equations with the simple and regular format. Due to without the derived functions in domain integral kernels, there have not to appear any singular domain integral kernels if the finite element approximations in domain are suitably selected, which is to yield some of convenience to ours coupling method. Furthermore the error estimates and a numerical result are given.We are concerned with the mathematical model of quadratic optimal control of some structural-acoustic coupling problems. The state equation is a coupling problem of Helmholtz equation in three dimensions and fourth-order plate equations on Winkle foundation in two dimensions. The boundary conditions satisfy the equilibrium condition and the compatibility condition. It is shown that the optimality system of equations are not only coupled via the Neumann boundary condition of the costate equation concerning the Helmholtz equation but also coupled via the loading of the costate equation concerning the plate equation.4. Optimal equilibrium solution for solving the Pareto optimal solution of the multi-objective control problemsSeveral new conceptions called optimal equilibrium value (or vector) and optimal equilibrium solution are introduced for solving the Pareto optimal solution of the multi-objective control problems. We proved that the set of all the optimal equilibrium solutions is a connected convex set consisting of some Pareto optimal solution under satisfying some convexity conditions. The optimal equilibrium solutions satisfy individual rationality and group rationality. One can obtain a kind of the optimal equilibrium solution, as a whole, superior to the conventional Nash' arbitration solution. We also prove that the optimal equilibrium solutions are equivalent to the solutions of a single objective control problem. By this result, a new and simple method is proposed for solving the Pareto optimal solution of the multi-objective control problems. Finally, Finally, a test problem is given to illustrate the efficiency of the optimal equilibrium solution.