Numerical schemes for the optimal input flow of a supply-chain

An innovative numerical technique is presented to adjust the inflow to a supply chain in order to achieve a desired outflow, reducing the costs of inventory, or the goods timing in warehouses. The supply chain is modelled by a conservation law for the density of processed parts coupled to an ODE for the queue buffer occupancy. The control problem is stated as the minimization of a cost functional J measuring the queue size and the quadratic difference between the outflow and the expected one. The main novelty is the extensive use of generalized tangent vectors to a piecewise constant control, which represent time shifts of discontinuity points. Such method allows convergence results and error estimates for an Upwind- Euler steepest descent algorithm, which is also tested by numerical simulations

On optimality conditions for optimal control problem in coefficients for Δp-Laplacian

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as L^infinity(Omega) coefficient of Delta_p-Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincaré Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair

On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the "noncoercivity trouble." Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions

Approximation of an Optimal Control Problem in the Coefficient for Variational Inequality with Anisotropic p-Laplacian

We study an optimal control problem for a variational inequality with the so-called anisotropic $p$-Laplacian in the principle part of this inequality. The coefficients of the anisotropic $p$-Laplacian, the matrix $A(x)$, we take as a control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the solutions $y\in K\subset W^{1,p}_0(\Omega)$ of the corresponding variational inequality. We show that the original problem is well-posed and derive existence of optimal pairs. Since the anisotropic $p$-Laplacian inherits the degeneracy with respect to unboundedness of the term $|(A(x)\nabla y,\nabla y)_{\mathbb{R}^N}|^{\frac{p-2}{2}}$, we introduce a two-parameter model for the relaxation of the original problem. Further we discuss the asymptotic behavior of relaxed solutions and show that some optimal pairs to the original problem can be attained by the solutions of two-parametric approximated optimal control problems

Qualitative Analysis of an Optimal Sparse Control Problem for Quasi-Linear Parabolic Equation with Variable Order of Nonlinearity

In this work, we study a sparse optimal control problem involving a quasilinear parabolic equation with variable order of nonlinearity as a state equation and with a pointwise control constraints. We show that in the case if the cost functional contains the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the compensated compactness technique and the consept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation

Unbounded Perturbations of the Generator Domain

Let X, U and Z be Banach spaces such that Z in X (with continuous and dense embedding), L : Z ->X be a closed linear operator and consider closed linear operators G, M : Z -> U. Putting conditions on G and M we show that the operator A = L with domain D(A) ={z∈Z: Gz=Mz} generates a C0- semigroup on X. Moreover, we give a variation of constants formula for the solution of an inhomogeneous problem. Several examples will be given, in particular heat equation with distributed unbounded delay at the boundary condition

Fictitious Controls and Approximation of an Optimal Control Problem for Perona-Malik Equation

We discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution ud 2 L2( ) and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero

On Optimization of a Highly Re-Entrant Production System

We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and quasi-pull constituents, and the position of SDP