Optimal control problem of the uncertain second‐order circuit based on first hitting criteria

First hitting criteria of a system are to initially achieve some performance indeces of the target state set. This paper primarily investigates the optimal control problem of the uncertain second‐order circuit based on first hitting criteria. First, considering time efficiency and different from the ordinary expected utility criterion over an infinite time horizon, two first hitting criteria which are reliability index and reliable time criteria are innovatively proposed. Second, assuming the circuit output voltage as an uncertain variable when the historical data is lacking, we better model the real circuit system with the uncertain second‐order differential equation which is essentially the uncertain fractional‐order differential equation. Then, based on the first hitting time theorem of the uncertain fractional‐order differential equation, the distribution function of the first hitting time under the second‐order circuit system is proposed and the uncertain second‐order circuit optimal control model (reliability index and reliable time‐based model) is transformed into corresponding crisp optimal problem. Lastly, analytic expressions of the optimal control for the reliability index model are obtained. Meanwhile, sufficient condition and guidance for parameters for the optimal solution of the reliable time‐based model are derived, and corresponding numerical examples are also given to demonstrate the fluctuation of our optimal solution for different parameters.     

带非局部低阶项非线性抛物型方程的时间最优控制

考虑一个带非局部低阶项非线性抛物型方程的时间最优控制问题．首先利用Schauder不动点定理证明了系统的适定性，然后利用Carleman不等式和Kakutani不动点定理证明了容许控制和最优控制的存在性，并且建立了时间最优控制的最大值原理．     

Snapshot location for POD in control of a linear heat equation

In the present work, we study the approximation of a distributed optimal control problem for a linear heat equation with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). We show that snapshot location for control problems is crucial in model reduction. For the determination of the time instances (snapshot locations) we utilize an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system. Finally, we present a numerical test to illustrate our approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of variational iteration method for Hamilton–Jacobi–Bellman equations

In this paper, we use the variational iteration method (VIM) for optimal control problems. First, optimal control problems are transferred to Hamilton–Jacobi–Bellman (HJB) equation as a nonlinear first order hyperbolic partial differential equation. Then, the basic VIM is applied to construct a nonlinear optimal feedback control law. By this method, the control and state variables can be approximated as a function of time. Also, the numerical value of the performance index is obtained readily. In view of the convergence of the method, some illustrative examples are presented to show the efficiency and reliability of the presented method.     

Time optimal sampled-data controls for the heat equation

In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.     

The Value Function of Singularly Perturbed Control Systems

The limit as ɛ→ 0 of the value function of a singularly perturbed optimal control problem is characterized. Under general conditions it is shown that limit value functions exist and solve in a viscosity sense a Hamilton—Jacobi equation. The Hamiltonian of this equation is generated by an infinite horizon optimization on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton—Jacobi equation are applicable in cases where the reduction of order, namely setting ɛ = 0 , does not yield an optimal behavior. Accepted 18 November 1999     

Residual-based a posteriori error control for Space-time finite element approximations of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.     

Limiting behavior of bang-bang controls for Sobolev problems

We consider the limiting behavior of optimal bang-bang controls as a family of Sobolev equations formally converges to a wave equation. The weak-starlimit of the sequence of bang-bang controls is an optimal control for the wave equation problem. The associated optimal states converge strongly and, for the optimal time problem, the optimal times converge to the optimal time for the wave equation.This work was supported in part by the National Science Foundation, Grant No. MCS-79-02037.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

Approximation of a class of optimal control problems with order of convergence estimates

An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation.     

Optimal double control problem for a PDE model of goodwill dynamics

We propose a new optimal control model of product goodwill in a segmented market where the state variable behaviour is described by a partial differential equation of the Lotka–Sharp–McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the marketing communication activities which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of heterogeneous electronic word of mouth. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we describe a numerical algorithm to find the optimal solution. Finally, we examine several examples on the optimal goodwill model and discover two types of marketing strategies.     

Second order sufficient conditions and sensitivity analysis for the optimal control of a container crane under state constraints

Stability and sensitivity analysis of parametric control problems has recently been elaborated for optimal control problems subject to pure state constraints. This paper illustrates the numerical aspects of sensitivity analysis for a complex practical example: the optimal control of a container crane with a state constraint on the vertical velocity. The multiple shooting method is used to determine a nominal solution satisfying first order necessary conditions. Second order sufficient conditions are checked by showing that an associated Riccati equation has a bounded solution. Sensitivity differentials of optimal solutions an computed with respect to variations in the swing angle     

Solving an optimal control problem for parabolic equations

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 90–98, 1986.     

Numerical solutions for optimal control of monodomain equations

We present the numerical solutions of optimality systems corresponding to optimal control problems governed by the mono-domain equations which are widely used for describing the electrical activity of the cardiac tissue. This mono-domain model is based on a parabolic equation and a system of stiff ordinary differential equations. The space discretization of the state variables and dual variables is done using piecewise linear finite elements and the time discretization is based on linearly implicit Runge-Kutta methods. The main goal of this work is to study the optimal control behavior of the electrical potentials under the influence of extracellular ionic currents as control variables. The numerical results presented are based on first and second order optimization methods. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results.     

A General Stochastic Maximum Principle for SDEs of Mean-field Type

We study the optimal control for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng’s-type stochastic maximum principle (Peng, S.: SIAM J. Control Optim. 2(4), 966–979, 1990) is derived, specifying the necessary conditions for optimality. This maximum principle differs from the classical one in the sense that here the first order adjoint equation turns out to be a linear mean-field backward SDE, while the second order adjoint equation remains the same as in Peng’s stochastic maximum principle.     

Properties and numerical testing of a parallel global optimization algorithm

In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman?CForchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville??s equation.     

Optimal Control Problem Governed by Semilinear Parabolic Equation and its Algorithm

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.