Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

Optimal control computation for nonlinear time-lag systems

In this paper, a computational scheme using the technique of control parameterization is developed for solving a class of optimal control problems involving nonlinear hereditary systems with linear control constraints. Several examples have been solved to test the efficiency of the technique.     

Finite element methods for optimal control problems governed by integral equations and integro-differential equations

In this paper, we analyze finite-element Galerkin discretizations for a class of constrained optimal control problems that are governed by Fredholm integral or integro-differential equations. The analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes.Grants, communicated-by lines, or other notes about the article will be placed here between rules. Such notes are optional.     

Optimal control problems for wave equations

The problem on minimizing a quadratic functional on trajectories of the wave equation is considered. We assume that the density of external forces is a control function. A control problem for a partial differential equation is reduced to a control problem for a countable system of ordinary differential equations by use of the Fourier method. The controllability problem for this countable system is considered. Conditions of the noncontrollability for some wave equations were obtained.     

Hybrid functions approach for optimal control of systems described by integro-differential equations

In this paper, a new numerical method for solving the optimal control of a class of systems described by integro-differential equations with quadratic performance index is presented. This optimization problem plays an important role in describing the dynamics of an elastic aircraft with allowance for non-steady flow past its profile. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of optimization problem to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Exact reachability for second-order integro-differential equations

In this Note we analyze a reachability problem for an integro-differential equation by using a harmonic analysis approach. To cite this article: P. Loreti, D. Sforza, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Optimal control for evolution equations with memory

In this paper, we investigate the existence and regularity of solutions for Bolza optimal control problems in infinite dimension governed by a class of semilinear evolution equations. Our results apply to systems exhibiting hereditary properties, as heat propagation in real conductors and isothermal viscoelasticity, described by equations with memory terms which account for the past history of the variables in play.     

Optimal control for semilinear stochastic evolution equations

Existence, uniqueness, continuous dependence with respect to controls and convergence in the probability of finite differences for controlled semilinear stochastic evolution equations, driven by continuous semimartingales, are considered under Lipschitz and monotone coefficients. The existence of discrete-optimal feedback controls for an associated optimization problem is proved.     

Sample path approximation for stochastic integro-differential equations

We consider the problem of mean square sample path convergence for a specialized stochastic difference scheme defined for stochastic integro-differential equations under the assumption of pathwise uniqueness property of solutions. We apply the method of weak convergence and the technique of Skorohod representation in the space of continuous trajectories     

An algorithm for solving Fredholm integro-differential equations

The aim of this paper is to present an efficient numerical procedure for solving linear second order Fredholm integro-differential equations. The scheme is based on B-spline collocation and cubature formulas. The analysis is accompanied by numerical examples. The results demonstrate reliability and efficiency of the proposed algorithm.      

Numerical methods for singular nonlinear integro-differential equations

For the numerical integration of singular nonlinear integro-differential equations we consider fractional linear multistep methods. We prove convergence of these methods and discuss their stability (as an extension of A-stability for stiff differential equations). Numerical experiments with the Basset equation are included.     

Regularity for anisotropic fully nonlinear integro-differential equations

We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality and the interior \(C^{1, \gamma }\) regularity, extending the results of Caffarelli and Silvestre (Comm Pure Appl Math 62:597–638, 2009) to the anisotropic case.     

Optimal control of continuity equations

An optimal control problem for the continuity equation is considered. The aim of a “controller” is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.     

A multipoint problem for partial integro-differential equations

We investigate a multipoint problem for a linear typeless partial differential operator with variable coefficients that is perturbed by a nonlinear integro-differential term. We establish conditions for the unique existence of a solution. We prove metric theorems on lower bounds of small denominators that arise in the course of investigation of the problem of solvability. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1155–1168, September, 1998.     

Collocation methods for second-order volterra integro-differential equations

We study the numerical solution of second-order Volterra integro-differential equations by means of collocation techniques in certain polynomial spline spaces. Suitable discretization of the resulting collocation equation yields implicit methods which may be viewed as extensions of m-stage implicit Runge-Kutta-Nyström methods for initial-value problems of second-order ordinary differential equations to second-order integro-differential equations. The attainable order of (local) convergence of these methods is analyzed in detail.     

Decomposition method for solving nonlinear integro-differential equations

This paper outlines a reliable strategy for solving nonlinear Volterra-Fredholm integro-differential equations. The modified form of Adomian decomposition method is found to be fast and accurate. Numerical examples are presented to illustrate the accuracy of the method.     

Least-squares Galerkin procedures for parabolic integro-differential equations

Two least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in H(div;Ω)×H¹(Ω) and (L²(Ω))²×L²(Ω), respectively.     

On spectral methods for Volterra-type integro-differential equations

This paper considers the spectral methods for a Volterra-type integro-differential equation. Firstly, the Volterra-type integro-differential equation is equivalently restated as two integral equations of the second kind. Secondly, a Legendre-collocation method is used to solve them. Then the error analysis is conducted based on the _L∞$L_{\infty }$-norm. In addition, numerical results are presented to confirm our analysis.