Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.     

Necessity of limiting co-state arcs in Bolza-type infinite horizon problem

We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstad’s idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev–Kryazhimskii-type formulae for co-state arcs. The connection between these formulae and limiting gradients of pay-off function at infinity is identified; several conditions under which it is possible to explicitly specify the co-state arc through those Aseev–Kryazhimskii-type formulae are found. For infinite horizon problem of Bolza type, an example is given to clarify the use of the Aseev–Kryazhimskii formula as an explicit expression of the co-state arc.     

Transversality condition and optimization of higher order ordinary differential inclusions

In the present paper, a Bolza problem of optimal control theory with a fixed time interval given by convex and nonconvex second-order differential inclusions (P_H) is studied. Our main goal is to derive sufficient optimality conditions for Cauchy problem of sth-order differential inclusions. The sufficient conditions including distinctive transversality condition are proved incorporating the Euler–Lagrange and Hamiltonian type inclusions. The basic concepts involved in obtaining optimality conditions are the locally adjoint mappings. Furthermore, the application of these results is demonstrated by solving the problems with third-order differential inclusions.     

Optimization of Second-Order Discrete Approximation Inclusions

The present article studies the approximation of the Bolza problem of optimal control theory with a fixed time interval given by convex and non-convex second-order differential inclusions (P _C ). Our main goal is to derive necessary and sufficient optimal conditions for a Cauchy problem of second-order discrete inclusions (P _D ). As a supplementary problem, discrete approximation problem (P _DA ) is considered. Necessary and sufficient conditions, including distinctive transversality, are proved by incorporating the Euler-Lagrange and Hamiltonian type of inclusions. The basic concept of obtaining optimal conditions is the locally adjoint mappings (LAM) and equivalence theorems, one of the most characteristic features of such approaches with the second-order differential inclusions that are peculiar to the presence of equivalence relations of LAMs. Furthermore, the application of these results are demonstrated by solving some non-convex problem with second-order discrete inclusions.     

Lipschitzianity of optimal trajectories for the Bolza optimal control problem

In this paper we investigate Lipschitz continuity of optimal solutions for the Bolza optimal control problem under Tonelli’s type growth condition. Such regularity being a consequence of normal necessary conditions for optimality, we propose new sufficient conditions for normality of state-constrained nonsmooth maximum principles for absolutely continuous optimal trajectories. Furthermore we show that for unconstrained problems any minimizing sequence of controls can be slightly modified to get a new minimizing sequence with nice boundedness properties. Finally, we provide a sufficient condition for Lipschitzianity of optimal trajectories for Bolza optimal control problems with end point constraints and extend a result from (J. Math. Anal. Appl. 143, 301–316, 1989) on Lipschitzianity of minimizers for a classical problem of the calculus of variations with discontinuous Lagrangian to the nonautonomous case.     

Terminality,normality, and transversality conditions

The maximum principle distinguishes between two phases of the optimal control problem. Some of the stated conditions are to be satisfied at points other than endpoints and some conditions are to be satisfied specifically at the endpoints. This paper utilizes the first set of conditions from the maximum principle to reexamine the second set. In the process, a new necessary condition to be satisfied at the endpoints is obtained. This condition is for certain cases easier to apply than the transversality conditions, yields additional information which may be of computational advantage, and lends itself quite naturally to an exposition of abnormal solutions. The relationship of the new condition to the transversality conditions and a discussion on normality are included. Several examples are given to illustrate the results.This research was supported in part by NASA under Grant NGR-03-002-224 and NSF Science Faculty Fellowship. The authors are indebted to Professors G. Leitmann, G. Basile, and E. Cliff for their helpful comments and suggestions.     

A Study on Abnormality in Variational and Optimal Control Problems

In this paper, a variational problem is considered with differential equality constraints over a variable interval. It is stressed that the abnormality is a local character of the admissible set; consequently, a definition of regularity related to the constraints characterizing the admissible set is given. Then, for the local minimum necessary conditions, a compact form equivalent to the well-known Euler equation and transversality condition is given. By exploiting this result and the previous definition of regularity, it is proved that nonregularity is a necessary and sufficient condition for an admissible solution to be an abnormal extremal. Then, a necessary and sufficient condition is given for an abnormal extremal to be weakly abnormal. The analysis of the abnormality is completed by considering the particular case of affine constraints over a fixed interval: in this case, the abnormality turns out to have a global character, so that it is possible to define an abnormal problem or a normal problem. The last section is devoted to the study of an optimal control problem characterized by differential constraints corresponding to the dynamics of a controlled process. The above general results are particularized to this problem, yielding a necessary and sufficient condition for an admissible solution to be an abnormal extremal. From this, a previously known result is recovered concerning the linearized system controllability as a sufficient condition to exclude the abnormality.     

Transversality condition for infinite-horizon problems

We present necessary conditions of optimality for an infinitehorizon optimal control problem. The transversality condition is derived with the help of stability theory and is formulated in terms of the Lyapunov exponents of solutions to the adjoint equation. A problem without an exponential factor in the integral functional is considered. Necessary and sufficient conditions of optimality are proved for linear quadratic problems with conelike control constraints.     

Sequential decision problems on isolated time domains

In this paper, we shall study the deterministic dynamic sequence problem on isolated time domains. After introducing the Euler equations and the transversality condition, we shall prove that the Euler equations and transversality condition are sufficient for the existence of the optimal solution. We shall also introduce the Bellman equation on isolated time scales. This equation will generalize the well-known Bellman equation in the theory of dynamic programming. As an application in financial economics, we shall optimize a sequence problem of growth model on isolated time domains.     

ON THE SUSTAINABLE GROWTH IN AN ECONOMY WITH PERFECTLY SUBSTITUTABLE EXHAUSTIBLE RESOURCES

Abstract A model of sustainable economic growth in an economy with two types of exhaustible resources is analyzed. The resources are assumed to be perfect substitutes with marginal rate of substitution varying over time. The optimal control framework is used to characterize the optimal paths under the maximin criterion. It is shown that the resource with increasing productivity is not used before the constant productivity resource is depleted. Afterwards the resource with an increasing productivity is asymptotically depleted as well. The results are based on an assumption that transversality conditions hold. A new sufficient condition for the transversality conditions is derived. Finally, an analogue of Hartwick’s rule for this non‐autonomous case is established.     

Optimal boundary displacement control of string vibrations with a nonlocal oddness condition of the first kind

We consider the problem of optimal boundary control by the displacement at left endpoint of a string in the case of a nonlocal oddness boundary condition of the first kind. We obtain a necessary and sufficient condition for the problem controllability under arbitrary initial and terminal conditions and construct a closed analytical form of the control itself under these conditions. In addition, we consider the problem of optimal boundary control by the displacement at one endpoint of the string for a given displacement mode at the other endpoint.     

Relaxation of a Bolza Problem Governed by a Time-Delay Sweeping Process

We study in an infinite dimensional Hilbert space a Bolza problem in which the dynamics are given by a time-delay perturbed sweeping process. This is a differential inclusion whose right-hand side involves a normal cone to a moving set, along with a time-delay perturbation. A relaxation result is established from which we deduce a sufficient condition ensuring the existence of an optimal solution.     

Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.     

Structure of minimizers in linear-quadratic Bolza problems of optimal control

In this paper, we study intersections of extremals in a linear-quadratic Bolza problem of optimal control. The structure of the inter-sections is described. We show that this structure implies the semipositive definiteness of the quadratic cost functional. In addition, we derive necessary and sufficient conditions for the existence of minimizers.     

Maximum principle in the problem of time optimal response with nonsmooth constraints : PMM vol. 40, n 6, 1976, pp. 1014–1023

The problem of optimal response [1, 2] with nonsmooth (generally speaking, nonfunctional) constraints imposed on the state variables is considered. This problem is used to illustrate the method of proving the necessary conditions of optimality in the problems of optimal control with phase constraints, based on constructive approximation of the initial problem with constraints by a sequence of problems of optimal control with constraint-free state variables. The variational analysis of the approximating problems is carried out by means of a purely algebraic method involving the formulas for the incremental growth of a functional [3, 4] and the theorems of separability of convex sets is not used.Using a passage to the limit, the convergence of the approximating problems to the initial problem with constraints is proved, and for general assumptions the necessary conditions of optimality resembling the Pontriagin maximum principle [1] are derived for the generalized solutions of the initial problem. The conditions of transversality are expressed, in the case of nonsmooth (nonfunctional) constraints by a novel concept of a cone conjugate to an arbitrary closed set of a finite-dimensional space. The concept generalizes the usual notions of the normal and the normal cone for the cases of smooth and convex manifolds.     

Calculus of variations on time scales: weak local piecewise Crd solutions with variable endpoints

A nonlinear calculus of variations problem on time scales with variable endpoints is considered. The space of functions employed is that of piecewise rd-continuously Δ-differentiable functions (C¹_prd). For this problem, the Euler-Lagrange equation, the transversality condition, and the accessory problem are derived as necessary conditions for weak local optimality. Assuming the coercivity of the second variation, a corresponding second order sufficiency criterion is established.     

A note on the dynamic liquidity trading problem with a mean-variance objective

This note studies the dynamic liquidity trader’s problem with a mean-variance objective function. Independent of the market impact functions and the market price dynamics, we provide a necessary and sufficient condition under which the dynamic programming equation (Bellman equation) can be extended to mean-variance objectives. Evaluation of this condition involves solving an optimization problem and taking variance of its optimal value. This computation may be difficult even when random disturbances in the market price dynamics follow a well-known distribution. To avoid this pitfall, we then provide some sufficient condition which can be assessed very easily.     

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified.     

Regularity of the Optimal Feedback and the Value Function in Convex Problems of Optimal Control

An optimal feedback mapping, leading to necessary and sufficient conditions for optimality in terms of a closed-loop differential inclusion, is derived in the setting of fully convex generalized problems of Bolza. Results are translated to the format of control problems with linear dynamics and convex costs. Properties of the feedback mapping, with focus on single-valuedness and continuity, are analyzed through those of the value function and of the Hamiltonian. Conditions guaranteeing differentiability of the value function are obtained through the analysis of its subdifferential as a maximal monotone operator and of the generalized Hamiltonian dynamics.     

Computation of uniform integral means and some similar characteristics of piecewise continuous functions

We obtain formulas for the computation of a certain asymptotic characteristic of piecewise continuous functions defined on the half-line and use it to state the well-known necessary and sufficient condition on a linear homogeneous differential system for the corresponding inhomogeneous system with arbitrary inhomogeneity whose characteristic exponent is nonpositive to have a solution with nonpositive characteristic exponent. We give a new form of this necessary and sufficient condition similar to the Perron-Maizel condition for the exponential dichotomy of the system.