211,554 research outputs found

    Numerical approximation of some time optimal control problems

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    International audienceIn this work we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems

    Shifting the closed-loop spectrum in the optimal linear quadratic regulator problem for hereditary systems

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    In the optimal linear quadratic regulator problem for finite dimensional systems, the method known as an alpha-shift can be used to produce a closed-loop system whose spectrum lies to the left of some specified vertical line; that is, a closed-loop system with a prescribed degree of stability. This paper treats the extension of the alpha-shift to hereditary systems. As infinite dimensions, the shift can be accomplished by adding alpha times the identity to the open-loop semigroup generator and then solving an optimal regulator problem. However, this approach does not work with a new approximation scheme for hereditary control problems recently developed by Kappel and Salamon. Since this scheme is among the best to date for the numerical solution of the linear regulator problem for hereditary systems, an alternative method for shifting the closed-loop spectrum is needed. An alpha-shift technique that can be used with the Kappel-Salamon approximation scheme is developed. Both the continuous-time and discrete-time problems are considered. A numerical example which demonstrates the feasibility of the method is included

    Least-squares finite elements for distributed optimal control problems

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    We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is inf\inf--sup\sup stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented

    Error estimates for a tree structure algorithm solving finite horizon control problems

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    In the Dynamic Programming approach to optimal control problems a crucial role is played by the value function that is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well known that this approach suffers of the "curse of dimensionality" and this limitation has reduced its practical in real world applications. Here we analyze a dynamic programming algorithm based on a tree structure. The tree is built by the time discrete dynamics avoiding in this way the use of a fixed space grid which is the bottleneck for high-dimensional problems, this also drops the projection on the grid in the approximation of the value function. We present some error estimates for a first order approximation based on the tree-structure algorithm. Moreover, we analyze a pruning technique for the tree to reduce the complexity and minimize the computational effort. Finally, we present some numerical tests

    Numerical controllability of the wave equation through primal methods and Carleman estimates

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    This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute an approximation of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not use in this work duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and of the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments

    Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

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    In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results

    Optimal control for nonlocal reaction-diffusion system describing calcium dynamics in cardiac cell

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    International audienceThe purpose of this paper is to introduce an optimal control for a nonlocal calcium dynamic model in a cardiac cell acting on ryanodine receptors. The optimal control problem is considered as a coupled nonlocal reaction-diffusion system with a transmission boundary condition covering the sarcoplasmic reticulum and cytosolic domain. We establish the well-posedness result of the adjoint problem using Faedo-Galerkin approximation, a priori estimates and compactness arguments. The numerical discretization of direct and adjoint problems is realized by using the implicit Euler method in time and the finite element for spatial discretization. Moreover, we obtain the stability result in the 2-norm for the direct and the adjoint discrete problems. Finally, in order to illustrate the control of our calcium dynamic model, we present some numerical experiments devoted to constant and nonlocal diffusions using the proposed numerical scheme

    Global algorithms for nonlinear discrete optimization and discrete-valued optimal control problems

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    Optimal control problems arise in many applications, such as in economics, finance, process engineering, and robotics. Some optimal control problems involve a control which takes values from a discrete set. These problems are known as discrete-valued optimal control problems. Most practical discrete-valued optimal control problems have multiple local minima and thus require global optimization methods to generate practically useful solutions. Due to the high complexity of these problems, metaheuristic based global optimization techniques are usually required.One of the more recent global optimization tools in the area of discrete optimization is known as the discrete filled function method. The basic idea of the discrete filled function method is as follows. We choose an initial point and then perform a local search to find an initial local minimizer. Then, we construct an auxiliary function, called a discrete filled function, at this local minimizer. By minimizing the filled function, either an improved local minimizer is found or one of the vertices of the constraint set is reached. Otherwise, the parameters of the filled function are adjusted. This process is repeated until no better local minimizer of the corresponding filled function is found. The final local minimizer is then taken as an approximation of the global minimizer.While the main aim of this thesis is to present a new computational methodfor solving discrete-valued optimal control problems, the initial focus is on solvingpurely discrete optimization problems. We identify several discrete filled functionstechniques in the literature and perform a critical review including comprehensive numerical tests. Once the best filled function method is identified, we propose and test several variations of the method with numerical examples.We then consider the task of determining near globally optimal solutions of discrete-valued optimal control problems. The main difficulty in solving the discrete-valued optimal control problems is that the control restraint set is discrete and hence not convex. Conventional computational optimal control techniques are designed for problems in which the control takes values in a connected set, such as an interval, and thus they cannot solve the problem directly. Furthermore, variable switching times are known to cause problems in the implementation of any numerical algorithm due to the variable location of discontinuities in the dynamics. Therefore, such problem cannot be solved using conventional computational approaches. We propose a time scaling transformation to overcome this difficulty, where a new discrete variable representing the switching sequence and a new variable controlling the switching times are introduced. The transformation results in an equivalent mixed discrete optimization problem. The transformed problemis then decomposed into a bi-level optimization problem, which is solved using a combination of an efficient discrete filled function method identified earlier and a computational optimal control technique based on the concept of control parameterization.To demonstrate the applicability of the proposed method, we solve two complex applied engineering problems involving a hybrid power system and a sensor scheduling task, respectively. Computational results indicate that this method is robust, reliable, and efficient. It can successfully identify a near-global solution for these complex applied optimization problems, despite the demonstrated presence of multiple local optima. In addition, we also compare the results obtained with other methods in the literature. Numerical results confirm that the proposed method yields significant improvements over those obtained by other methods
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