Singular Arcs in the Optimal Evasion Against a Proportional Navigation Vehicle

The two-dimensional optimal evasion problem against a proportional navigation pursuer is analyzed using a nonlinear model. The velocities of both players have constant modulus, but change in direction. The problem is to determine the time-minimum trajectory (disengagement) or time-maximum trajectory (evasion) of the evader while moving from the assigned initial conditions to the final conditions. A maximum principle procedure allows one to reduce the optimal control problem to the phase portrait analysis of a system of two differential equations. The qualitative features of the optimal process are determined.     

Maximum Principle for State-Constrained Control of Some Semilinear Parabolic Differential Equations

This paper is concerned with the Pontryagin maximum principle for optimal control problems of some semilinear equations, which may be governed by nonmonotone operators and in particular might be singular. Some state constraints are considered.     

More on optimal control with paths lying on a corner

This paper extends the Pontryagin maximum principle to allow for a finite number of corners, i.e., allowing for any finite number of discontinuities of the first derivatives with respect to the state variables. These corners are shown to raise the same computational difficulties caused by state constraints.     

Optimal Control of Magnetohydrodynamic Equations with State Constraint

This work is concerned with the maximum principle for optimal control problem governed by magnetohydrodynamic equations, which describe the motion of a viscous incompressible conducting fluid in a magnetic field and consist of a subtle coupling of the Navier-Stokes equation of viscous incompressible fluid flow and the Maxwell equation of electromagnetic field. An integral type state constraint is considered.     

Optimal portfolio strategies benchmarking the stock market

The paper investigates the impact of adding a shortfall risk constraint to the problem of a portfolio manager who wishes to maximize his utility from the portfolios terminal wealth. Since portfolio managers are often evaluated relative to benchmarks which depend on the stock market we capture risk management considerations by allowing a prespecified risk of falling short such a benchmark. This risk is measured by the expected loss in utility. Using the Black–Scholes model of a complete financial market and applying martingale methods, explicit analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Numerical examples illustrate the analytic results.     

Optimal control problems with state constraints for semilinear distributed-parameter systems

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.This research was partly supported by the National Science Foundation under Grant DMS-92-21819.     

Optimal investment in the reclamation of eutrophic water bodies

Eutrophication, i.e., the abnormal growth of phytoplankton, is considered in this note, which focuses on the optimal treatment of eutrophic water bodies. The issue is addressed by the use of a nonlinear model where phytoplankton and the number of wastewater treatment plants in operation are the state variables. The decision maker is a governmental agency which has to define the time pattern of investment in new plants so as to minimize the present value of environmental and treatment costs. The optimal solution is shown to have the following features. First, the optimal size for the wastewater treatment system is attained in minimum time. Subsequently, investment replaces wornout treatment plants, and phytoplankton adjusts asymptotically to its optimal equilibrium value.This work was supported by Centro Teoria dei Sistemi—CNR, Milano and by Fondazione ENI E. Mattei.     

Characterizations of Optimal Trajectories for Nonconvex Control Systems under State Constraints

We consider an optimal control problem for a nonconvex control system under state constraints and the associated value function, which in general is not differentiable. We provide some characterizations of optimal trajectories using contingent derivatives. For this aim, we derive a costate satisfying the adjoint equation, the maximum principle, and a transversality condition linked to the superdifferential of the value function.Communicated by F. ZirilliThis paper is dedicated by the author to her children.     

Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.The author thanks the referees for careful reading and helpful suggestions and comments.     

Necessary extremum conditions in the optimal control problem with state constraints

The optimal control problem with state constraints is examined. An alternative to the available approaches to the study of this problem is proposed. The maximum principle and second-order necessary conditions are proved.     

Optimal prevention strategies in the classical risk model

In this paper, we propose and study a first risk model in which the insurer may invest into a prevention plan which decreases claim intensity. We determine the optimal prevention investment for different risk indicators. In particular, we show that the prevention amount minimizing the ruin probability maximizes the adjustment coefficient in the classical ruin model with prevention, as well as the expected dividends until ruin in the model with dividends. We also show that the optimal prevention strategy is different if one aims at maximizing the average surplus at a fixed time horizon. A sensitivity analysis is carried out. We also prove that our results can be extended to the case where prevention starts to work only after a minimum prevention level threshold.     

Optimal control problems with applications to the elimination of substrate diffusing and convecting through immobilized enzyme

Some optimization problems concerning a substrate in a fluid are considered. The concentration of the substrate is affected by diffusion, convection, and elimination by enzymes, and the problem is to find the optimal distribution of enzymes. In this paper, the rate of elimination and the transmission coefficient are optimized. Mathematically, these problems are optimal control problems, and they are analyzed by means of Pontryagin's maximum principle.     

Representation of the Lagrange Multipliers for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The structure of the optimal control problem under consideration is exploited and special emphasis is laid on the representation of the Lagrange multipliers resulting from the necessary conditions for infinite optimization problems.The author thanks the referees for careful reading and helpful suggestions and comments.     

Optimal solutions to differential inclusions in presence of state constraints

Necessary conditions in terms of the Hamiltonian are given for optimal solutions to the differential inclusion problem when state constraints are present. This result extends a result of Clarke for the unconstrained problem. The data are nonsmooth, nonlinear, nonconvex. The method incorporates the state constraint in the cost functional as a penalty term for a sequence of unconstrained problems that approximate our problem. An application of Ekeland's variational principle, the known necessary conditions for the auxiliary problems, and a limiting process provide the necessary conditions.     

On the Stochastic Maximum Principle in Optimal Control of Degenerate Diffusions with Lipschitz Coefficients

We establish a stochastic maximum principle in optimal control of a general class of degenerate diffusion processes with global Lipschitz coefficients, generalizing the existing results on stochastic control of diffusion processes. We use distributional derivatives of the coefficients and the Bouleau Hirsh flow property, in order to define the adjoint process on an extension of the initial probability space. This work is partially supported by MENA Swedish Algerian Research Partnership Program (348-2002-6874) and by French Algerian Cooperation, Accord Programme Tassili, 07 MDU 0705.     

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.     

Optimal dividend-financing strategies in a dual risk model with time-inconsistent preferences

In this paper, we consider an optimal dividend-financing problem for a company whose capital reserve is described by the dual of classical risk model. We assume that the manager of the company has time-inconsistent preferences, which are described by a quasi-hyperbolic discount function, and that financing is permitted to prevent the company from going bankrupt. The manager’s objective is to maximize the expected cumulative dividend payments minus financing costs. We solve the optimization problems for a naive manager and a sophisticated manager, and obtain explicit solutions for both managers. Our results show that the manager with time-inconsistent preferences tends to pay out dividends earlier. We also present some economic implications and sensitivity analysis for our results.     

Proximal normal analysis approach to optimal control problems in infinite-dimensional spaces

In this paper, we extend the Pontryagin maximum principle and the transversality conditions to a class of optimal control problems for an evolution system of parabolic type through the analysis of proximal normals to the epigraph of suitable value functions. The paper extends previous results of the same authors to nonconvex target situations.This work was supported by MURST of Italy, Fondi 40%, Equazioni di Evoluzione ed Applicazioni Fisico-Matematiche, and Fondi 60%, University of Bari and University of Calabria.