On the time optimization problem for two-dimensional control systems

We consider a time optimization problem for two-dimensional nonlinear control systems. For a constraint for the control vector u we take a convex polygon U. We obtain effective sufficient conditions under which the time-optimal control is equivalent in the Lebesgue sense to a piecewise constant control that has finitely many points of discontinuity and ranges in the set of vertices of U. We give examples illustrating the results.     

Local solvability of control systems with mixed constraints

We analyze the local solvability of a control system with mixed constraints and control constraints. We obtain a sufficient condition for the local solvability of the control system under the assumption of smoothness and 2-regularity of the mapping g specifying the mixed constraints. For the case in which the mapping g is not smooth, a sufficient condition for the local solvability is obtained under the assumption of coverability of g.     

PDE-constrained control using Femlab – Control of the Navier–Stokes equations

We show how the software Femlab can be used to solve PDE-constrained optimal control problems. We give a general formulation for such kind of problems and derive the adjoint equation and optimality system. Then these preliminaries are specified for the stationary Navier–Stokes equations with distributed and boundary control. The main steps to define and solve a PDE with Femlab are described. We describe how the adjoint system can be implemented, and how the optimality system can be used by Femlab’s built-in functions. Special crucial topics concerning efficiency are discussed. Examples with distributed and boundary control for different type of cost functionals in 2 and 3 space dimensions are presented.     

Approximate Optimal Control and Stability of Nonlinear Finite- and Infinite-Dimensional Systems

We consider first nonlinear systems of the formx=A(x)x+B(x)u together with a standard quadratic cost functional and replace the system by a sequence of time-varying approximations for which the optimal control problem can be solved explicitly. We then show that the sequence converges. Although it may not converge to a global optimal control of the nonlinear system, we also consider a similar approximation sequence for the equation given by the necessary conditions of the maximum principle and we shall see that the first method gives solutions very close to the optimal solution in many cases. We shall also extend the results to parabolic PDEs which can be written in the above form on some Hilbert space.     

Control of thermoelastic stresses in a cylinder in the case of a two-dimensional nonaxisymmetric temperature field

We obtain new functional relations among the components of thermal stresses, and also their dependence on the temperature field. Using these relations we find solutions of a control problem that we pose. We state necessary conditions for the existence of the required optimal control which must be satisfied by the given distribution function of the control of thermal stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 56–60.     

Analytic form of optimal boundary controls of transverse vibrations of a rod consisting of several parts with different densities and elasticities but with the same impedances

We consider the problem of optimal boundary displacement control (control minimizing the boundary energy functional) of an elastic rod consisting of several parts with different densities and elasticities but with the same impedances. We obtain a closed-form expression for the optimal control bringing the rod from an arbitrarily given initial state to an arbitrarily given terminal state in a given time T.     

Structural-information modeling of administrative control processes

We consider the fundamental principles of structural-information modeling of administrative control processes. We present a formal model for investigation of these processes, and we formulate criteria for efficiency of the facilities used for automated administrative control. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 135–140, 1993.     

Optimal damping of perturbations in a system with unknown initial conditions

We consider the perturbation damping problem for a system in which, along with an external perturbation bounded in the L ₂-norm, there is an initial perturbation caused by unknown nonzero initial conditions. We state necessary and sufficient conditions for the existence of an optimal control law minimizing the maximum L ₂-norm of the system output for all L ₂-bounded external perturbations and bounded initial states and synthesize this control law.     

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L ¹-norm and with order 1/2 w.r.t. the L ²-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.     

Example of a linear abnormal optimal control problem

We give an example of a linear optimal control problem for which the Pontryagin maximum principle holds on the optimal control only for λ ⁰ = 0.     

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H^∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L^p(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω.     

Nonlinear Goursat-Darboux control problem: Conditions for the preservation of global solvability

We consider the nonlinear vector Goursat-Darboux problem of general form with a control on the right-hand side in the differential equation. We consider the case in which its solution necessarily belongs to the class of functions with p-integrable (for some p > 1) mixed and first partial derivatives. In this connection, we use the fact that the first derivatives of such a solution are elements of spaces with a mixed norm; these spaces are much narrower than L _p . Sufficient conditions for the preservation of the global solvability of the problem under control perturbations are proved.     

Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

Optimal control of unilateral obstacle problem with a source term

We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in H². We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions. The author is grateful to Prof. M. Bergounioux for her instructive suggestions.     

The shuttle dispatch problem with compound Poisson arrivals: Controls at two terminals

We consider the control of an infinite capacity shuttle which transports passengers between two terminals. The passengers arrive at each terminal according to a compound Poisson process and the travel time from one terminal to the other is a random variable following an arbitrary distribution. The following control limit policy is considered: dispatch the shuttle at terminali, at the instant that the total number of passengers waiting at terminali reaches or exceeds a predetermined control limitm _i . The objective of this paper is to obtain the mean waiting time of an arbitrary passenger at each terminal for given control valuesm ₁ andm ₂. We also discuss a search procedure to obtain the optimal control values which minimize the total expected cost per unit time under a linear cost structure.     

Admission control with advance reservations in simple networks

In the admission control problem we are given a network and a set of connection requests, each of which is associated with a path, a time interval, a bandwidth requirement, and a weight. A feasible schedule is a set of connection requests such that at any given time, the total bandwidth requirement on every link in the network is at most 1. Our goal is to find a feasible schedule with maximum total weight.We consider the admission control problem in two simple topologies: the line and the tree. We present a 12c-approximation algorithm for the line topology, where c is the maximum number of requests on a link at some time instance. This result implies a 12c-approximation algorithm for the rectangle packing problem, where c is the maximum number of rectangles that cover simultaneously a point in the plane. We also present an O(logt)-approximation algorithm for the tree topology, where t is the size of the tree. We consider the loss minimization version of the admission control problem in which the goal is to minimize the weight of unscheduled requests. We present a c-approximation algorithm for loss minimization problem in the tree topology. This result is based on an approximation algorithm for a generalization of set cover, in which each element has a covering requirement, and each set has a covering potential. The approximation ratio of this algorithm is Δ, where Δ is the maximum number of sets that contain the same element.     

Distributed control in a class of nonlocal boundary-value problems

We study optimal control problems with a quadratic optimization criterion. The state of the system is described by the solution of an operator-differential equation under nonlocal boundary conditions; the control is on the righth- and side of the equation. We prove existence and uniqueness theorems for the optimal control.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 50–54.     

Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

We consider a steady-state heat conduction problem P _α with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ ₁ of the boundary of a given bounded domain in R ⁿ . We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op _α and its corresponding system u_ g_ op _α α and adjoint p_ g_ op _α α states for each α are strongly convergent to g _op , u_ g _op and p _ g _op , respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control, and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation with a different boundary condition on the portion Γ ₁ . We use the fixed point and elliptic variational inequality theories.     

Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

We consider a steady-state heat conduction problem P _α with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ ₁ of the boundary of a given bounded domain in R ⁿ . We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op _α and its corresponding system u_ g_ op _α α and adjoint p_ g_ op _α α states for each α are strongly convergent to g _op , u_ g _op and p _ g _op , respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control, and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation with a different boundary condition on the portion Γ ₁ . We use the fixed point and elliptic variational inequality theories.     

Optimal boundary control of the wave equation with pointwise control constraints

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L ^∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L ²-norm has a solution that we present in this paper.