Let. (x)over dot = g(t, x(t), u(t)) be the governing equation of an optimalcontrol problem with two-point boundary conditions h(0)(x(a)) + h(1)(x(b)) = 0, where x:[a, b] --> R-n is continuous, u:[a, b] --> Rk-n...
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Let. (x)over dot = g(t, x(t), u(t)) be the governing equation of an optimalcontrol problem with two-point boundary conditions h(0)(x(a)) + h(1)(x(b)) = 0, where x:[a, b] --> R-n is continuous, u:[a, b] --> Rk-n is piecewise continuous and left continuous, h(0), h(1) : R-n --> R-q are continuously differentiable, and g:[a, b] x R-k --> R-n is continuous. The paper finds functions xi(i) is an element ofC(1)([a, b] x R-n) such that (x(t), u(t)) is a solution of the governing equation if and only if integral(a)(b) [(partial derivativexi(i)/partial derivativex)g + partial derivativexi(i)/partial derivativet]dt = 0, i = 1, 2, 3,....
In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated. Piecewise linear polynomials are used to approximate the state and adjoi...
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In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated. Piecewise linear polynomials are used to approximate the state and adjoint state, while the control is discretized by variational discretization method. A priori error estimates for the semi-discrete approximations of the state, adjoint state and control are derived. Furthermore, we also discuss the fully discrete scheme for the controlproblems. A finite difference method developed in Lin and Xu (2007) is used to discretize the time fractional derivative. Fully discrete first order optimality condition is developed based on 'first discretize, then optimize' approach. The stability and truncation error of the fully discrete scheme are analyzed. Numerical example is given to illustrate the theoretical findings. (C) 2015 Elsevier Ltd. All rights reserved.
Various first-order and second-order sufficient conditions of optimality for nonlinear optimal control problems with delayed argument are formulated. The functions involved are not required to be convex. Second-order ...
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Various first-order and second-order sufficient conditions of optimality for nonlinear optimal control problems with delayed argument are formulated. The functions involved are not required to be convex. Second-order sufficient conditions are shown to be related to the existence of solutions of a Riccati-type matrix differential inequality. Their relation with the second variation is discussed.
This paper studies variational discretization for the optimalcontrol problem governed by parabolic equations with control constraints. First of all, the authors derive a priori error estimates where｜｜｜u - Uh｜｜｜...
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This paper studies variational discretization for the optimalcontrol problem governed by parabolic equations with control constraints. First of all, the authors derive a priori error estimates where｜｜｜u - Uh｜｜｜L∞（J;L2（Ω）） = O（h2 ＋ k）. It is much better than a priori error estimates of standard finite element and backward Euler method where ｜｜｜u- Uh｜｜｜L∞（J;L2（Ω）） = O（h ＋ k）. Secondly, the authors obtain a posteriori error estimates of residual type. Finally, the authors present some numerical algorithms for the optimalcontrol problem and do some numerical experiments to illustrate their theoretical results.
In this paper, the differential transform method (DTM) is applied for solving time-invariant state-feedback controlproblems. The optimal equations are obtained using the Pontryagin's maximum principle (PMP) and B...
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In this paper, the differential transform method (DTM) is applied for solving time-invariant state-feedback controlproblems. The optimal equations are obtained using the Pontryagin's maximum principle (PMP) and Bellman's Dynamic Programming. We present the closed-loop optimalcontrol of linear plants with quadratic performance index. The results reveal that the proposed methods are very effective and simple. Comparisons are made between the results of two proposed methods and the exact solutions.
We investigate the possibility of describing the limit problem of a sequence of optimal control problems (P)((b n)), each of which is characterized by the presence of a time dependent vector valued coefficient b(n) = ...
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We investigate the possibility of describing the limit problem of a sequence of optimal control problems (P)((b n)), each of which is characterized by the presence of a time dependent vector valued coefficient b(n) = (b(n 1),..., b(n M)). The notion of limit problem is intended in the sense of Gamma -convergence, which, roughly speaking, prescribes the convergence of both the minimizers and the in mum values. Due to the type of growth involved in each problem (P)((b n)) the ( weak) limit of the functions (b(n 1)(2),...,b(n M)(2))-beside the limit (b(1),...,b(M)) of the (b(n 1),...,b(n M)) is crucial for the description of the limit problem. Of course, since the b(n) are L-2 maps, the limit of the (b(n 1)(2),...,b(n M)(2)) may well be a ( vector valued) measure mu = (mu (1),...,mu (M)). It happens that when the problems (P) (b(n)) enjoy a certain commutativity property, then the pair (b,mu) is sufficient to characterize the limit problem. This is no longer true when the commutativity property is not in force. Indeed, we construct two sequences of problems (P)((b n)) and (P)(((b) over bar n)) which are equal except for the coefficient b(n) ((.)) and (b) over tilde (n)((.)), respectively. Moreover, both the sequences (b(n), b(n)(2)) and ((b) over tilde (n), (b) over tilde (2)(n)) converge to the same pair (b,mu). However, the infimum values of the problems (P)((bn)) tend to a value which is different from the limit of the infimum values of the (P)(((b) over tilde n)). This means that the mere information contained in the pair (b,mu) is not sufficient to characterize the limit problem. We overcome this drawback by embedding the problems in a more general setting where limit problems can be characterized by triples of functions (B0, B, y) with B0 greater than or equal to0.
In this paper, the homotopy analysis method (HAM) is employed to solve the linear optimal control problems (OCPs), which have a quadratic performance index. The study examines the application of the homotopy analysis ...
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In this paper, the homotopy analysis method (HAM) is employed to solve the linear optimal control problems (OCPs), which have a quadratic performance index. The study examines the application of the homotopy analysis method in obtaining the solution of equations that have previously been obtained using the Pontryagin's maximum principle (PMP). The HAM approach is also applied in obtaining the solution of the matrix Riccati equation. Numerical results are presented for several test examples involving scalar and 2nd-order systems to demonstrate the applicability and efficiency of the method. (C) 2013 Elsevier Inc. All rights reserved.
In this article, line-up competition algorithm (LCA), a brand-new nonlinear programming method based on the principle of evolution, is applied to solve time-delay optimal control problems (TDOCPs). The problems are fi...
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In this article, line-up competition algorithm (LCA), a brand-new nonlinear programming method based on the principle of evolution, is applied to solve time-delay optimal control problems (TDOCPs). The problems are first discretized based on the concept of control vector parametrization, and then solved by LCA. Since the delay differential equations are directly integrated without using the auxiliary procedures Such as model conversion and data interpolation, most TDOCPs can be solved very conveniently under such solution framework. Meanwhile, a more efficient sampling strategy is adopted to promote the too slow convergence of the basic LCA. By solving six typical examples, including five pure mathematical problems and one chemical engineering problem, the modified LCA demonstrates a robust and efficient property in optimizing time-delay unsteady systems. (C) 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
A neural network scheme based on quintic B-spline functions of optimality systems arising from linear parabolic optimal control problems is discussed. It is also shown that the proposed neural network model is stable ...
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A neural network scheme based on quintic B-spline functions of optimality systems arising from linear parabolic optimal control problems is discussed. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to the optimal solution of the original problem. Theoretical and experimental results show the advantages of B-spline functions and demonstrate that the suggested scheme is able to efficiently solve constrained optimal control problems. The resulting scheme also shows that robustness with respect to changes of the value of v, the weight of the cost of the control, is sufficiently small.
A new approach to finding the approximate solution of distributed-order fractional optimal control problems (D-O FOCPs) is proposed. This method is based on Fibonacci wavelets (FWs). We present a new Riemann-Liouville...
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A new approach to finding the approximate solution of distributed-order fractional optimal control problems (D-O FOCPs) is proposed. This method is based on Fibonacci wavelets (FWs). We present a new Riemann-Liouville operational matrix for FWs using the hypergeometric function. Using this, an operational matrix of the distributed-order fractional derivative is presented. Implementing the mentioned operational matrix with the help of the Gauss-Legendre numerical integration, the problem converts to a system of algebraic equations. Error analysis is proposed. Finally, the validation of the present technique is checked by solving some numerical examples.
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