椭圆型分布参数最优控制问题的数值算法
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摘要
分布参数最优控制问题是用偏微分方程,或偏微分积分方程,或偏微分方程与常微分方程的耦合方程来描述的无限维控制系统。该类问题已广泛应用于航天技术、土木工程、生态系统、社会系统等工程技术领域。分布参数最优控制问题的数值算法研究是该领域的重要分支之一。目前,其数值算法主要是有限差分算法、多重网格方法、有限元算法和混合有限元方法等。本文主要研究椭圆型分布参数最优控制问题的混合有限元方法、边界元方法、有限元-边界元耦合方法以及多目标问题的最优均衡解方法。给出这些算法的理论证明、算法步骤和误差分析,并分别用算例验证这些算法。这些算法要么是目前算法中没有的,要么改进了目前已有的算法。本文主要成果如下:
1.混合有限元算法
给出双调和状态方程控制的边界最优控制问题的梯度型-混合有限元迭代算法。引入具有固支边界条件的共态方程,建立由状态函数和共态函数构成的最优性方程组。通过最优性系统发现,在最优性条件下边界上对应于共态方程的涡流函数恰好等于边界控制函数的某一倍数,这样自动产生了合适的梯度。基于此梯度建立梯度型混合有限元近似算法。该算法中共态方程涡流函数在边界上的迹映射至关重要。此外,给出了算法的局部误差估计,并用算例验证了算法。
2.边界元算法
在一定凸性条件下,建立了Helmholtz型状态方程控制的最优控制问题的最优性方程组。利用一阶必要性条件给出连续情形下的共轭梯度算法(CGM)。引进二维Helmholtz方程的高阶基本解序列,得到基于多重替换方法(MRM)的离散型共轭梯度-边界元迭代算法(CGM+MRBEM)。该方法优于现有的方法,因为充分利用了边界元方法的主要优势(实现降维)。此外,给出了CGM+MRBEM算法的局部误差估计,并以算例验证了算法的有效性。
引进二维四阶Winkle地基上板弯曲问题的高阶基本解序列,给出四阶状态方程控制的分布型最优控制问题的共轭梯度-多重替换边界元算法(CGM+MRBEM)。基于算法的稳定性,得到其局部H~2模误差估计、L~2模误差估计以及L~∞模误差估计,并用算例验证之。
3.有限元-边界元耦合算法
提出求解二维稳态固体燃烧问题的分布型最优控制的有限元-边界元耦合算法,其状态方程为具有非线性指数项的边值问题。得到最优控制问题的一阶必要性条件。引进二维Laplace方程的基本解,导出最优性系统的边界积分方程。将非线性指数项作为一个整体构造区域函数的有限元近似构造一个格式简单规范的非线性方程组。由于区域积分核中不包含导数项,因此,如果能选择适当的有限元近似,区域积分中不会出现奇异积分,这给耦合方法带来极大的方便。最后,讨论耦合方法的误差估计,并给出算例。
建立结构声学耦合系统的二次最优控制数学模型,其状态方程为满足相容边界条件和平衡边界条件的二维四阶Winkle地基上的板弯曲问题和三维Helmholtz方程的耦合问题。最优性方程组是耦合型的,其耦合不仅是三维Helmholtz方程对应的共态方程的Neumann边界条件的耦合,而且是弹性板方程对应的共态方程的荷载项之间的耦合。
4.求解多目标问题Pareto最优解的最优均衡解方法
引进求解多目标问题Pareto最优解的最优均衡值(或向量)和最优均衡解等概念,在某种凸性条件下,证明了最优均衡解集合是某些Pareto最优解的连通凸集。最优均衡解同时满足个体合理性和群体合理性。从整体而言,可得到一类优于Nash仲裁解的最优均衡解。证明了求解最优均衡解等价于求解一个单目标问题。最优均衡解方法是求解多目标问题Pareto最优解的新的、简单方法。最后,给出算例验证最优均衡解的有效性。
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 method
A 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 method
An 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 element
The 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 problems
Several 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.
1.混合有限元算法
给出双调和状态方程控制的边界最优控制问题的梯度型-混合有限元迭代算法。引入具有固支边界条件的共态方程,建立由状态函数和共态函数构成的最优性方程组。通过最优性系统发现,在最优性条件下边界上对应于共态方程的涡流函数恰好等于边界控制函数的某一倍数,这样自动产生了合适的梯度。基于此梯度建立梯度型混合有限元近似算法。该算法中共态方程涡流函数在边界上的迹映射至关重要。此外,给出了算法的局部误差估计,并用算例验证了算法。
2.边界元算法
在一定凸性条件下,建立了Helmholtz型状态方程控制的最优控制问题的最优性方程组。利用一阶必要性条件给出连续情形下的共轭梯度算法(CGM)。引进二维Helmholtz方程的高阶基本解序列,得到基于多重替换方法(MRM)的离散型共轭梯度-边界元迭代算法(CGM+MRBEM)。该方法优于现有的方法,因为充分利用了边界元方法的主要优势(实现降维)。此外,给出了CGM+MRBEM算法的局部误差估计,并以算例验证了算法的有效性。
引进二维四阶Winkle地基上板弯曲问题的高阶基本解序列,给出四阶状态方程控制的分布型最优控制问题的共轭梯度-多重替换边界元算法(CGM+MRBEM)。基于算法的稳定性,得到其局部H~2模误差估计、L~2模误差估计以及L~∞模误差估计,并用算例验证之。
3.有限元-边界元耦合算法
提出求解二维稳态固体燃烧问题的分布型最优控制的有限元-边界元耦合算法,其状态方程为具有非线性指数项的边值问题。得到最优控制问题的一阶必要性条件。引进二维Laplace方程的基本解,导出最优性系统的边界积分方程。将非线性指数项作为一个整体构造区域函数的有限元近似构造一个格式简单规范的非线性方程组。由于区域积分核中不包含导数项,因此,如果能选择适当的有限元近似,区域积分中不会出现奇异积分,这给耦合方法带来极大的方便。最后,讨论耦合方法的误差估计,并给出算例。
建立结构声学耦合系统的二次最优控制数学模型,其状态方程为满足相容边界条件和平衡边界条件的二维四阶Winkle地基上的板弯曲问题和三维Helmholtz方程的耦合问题。最优性方程组是耦合型的,其耦合不仅是三维Helmholtz方程对应的共态方程的Neumann边界条件的耦合,而且是弹性板方程对应的共态方程的荷载项之间的耦合。
4.求解多目标问题Pareto最优解的最优均衡解方法
引进求解多目标问题Pareto最优解的最优均衡值(或向量)和最优均衡解等概念,在某种凸性条件下,证明了最优均衡解集合是某些Pareto最优解的连通凸集。最优均衡解同时满足个体合理性和群体合理性。从整体而言,可得到一类优于Nash仲裁解的最优均衡解。证明了求解最优均衡解等价于求解一个单目标问题。最优均衡解方法是求解多目标问题Pareto最优解的新的、简单方法。最后,给出算例验证最优均衡解的有效性。
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 method
A 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 method
An 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 element
The 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 problems
Several 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.
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