Some Facts about the Ramsey Model

In modeling the dynamics of capital, the Ramsey equation coupled with the Cobb–Douglas production function is reduced to a linear differential equation by means of the Bernoulli substitution. This equation is used in the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. This result enriches our understanding of the model analysis in the optimal control framework.     

Approximate Penalty Method in Optimal Control Problems for Nonsmooth Singular Systems

We consider an abstract optimal control problem with additional constraints and nonsmooth terms, but without the requirement that both the state equation on the set of admissible controls and the extremum problem be solvable. We use the approximate penalty method proposed here to find an approximate (in the weak sense) solution of the problem. As an example, we consider the optimal control problem for a singular nonlinear elliptic type equation.     

Relaxation in nonconvex optimal control problems described by fractional differential equations

We consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control. A relaxation problem is treated along with the original problem. It is proved that, under general assumptions, the relaxation problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies simultaneously.     

Optimization for nonlinear hyperbolic equations without the uniqueness theorem for a solution of the boundary-value problem

We consider the optimal control problem for a system governed by a nonlinear hyperbolic equation without any constraints on the parameter of nonlinearity. No uniqueness theorem is established for a solution to this problem. The control-state mapping of this system is not Gateaux differentiable. We study an approximate solution of the optimal control problem by means of the penalty method.     

Asymptotics of the solution in a problem of optimal boundary control of a flow through a part of the boundary

We consider a problem of optimal control through a part of the boundary of solutions to an elliptic equation in a bounded domain with smooth boundary with a small parameter at the Laplace operator and integral constraints on the control. A complete asymptotic expansion of the solution to this problems in powers of the small parameter is constructed.     

A complete asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with geometric constraints

We consider an optimal control problem for solutions of a boundary value problem on an interval for a second-order ordinary differential equation with a small parameter at the second derivative. The control is scalar and is subject to geometric constraints. Expansions of a solution to this problem up to any power of the small parameter are constructed and justified.     

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.     

Optimal control problem with an integral equation as the control object

We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.     

On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function

We consider a class of infinite-horizon optimal control problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of that kind the initial state is fixed, no constraints are imposed on the behavior of the admissible trajectories at infinity, and the objective functional is given by a discounted improper integral. Earlier, for such problems, S.M. Aseev and A.V. Kryazhimskiy in 2004–2007 and jointly with the author in 2012 developed a method of finite-horizon approximations and obtained variants of the Pontryagin maximum principle that guarantee normality of the problem and contain an explicit formula for the adjoint variable. In the present paper those results are extended to a more general situation where the instantaneous utility function need not be locally bounded from below. As an important illustrative example, we carry out a rigorous mathematical investigation of the transitional dynamics in the neoclassical model of optimal economic growth.     

Discretization of elliptic control problems with time dependent parameters

We consider linear-quadratic problems of optimal control with an elliptic state equation and control constraints. For a discretization of the state equation by the method of Finite Differences and a piecewise approximation of the control we develop error estimates for the solution of the discrete problem and, based on the optimality conditions, we construct a new feasible control for which we derive error estimates of quadratic order.     

Method of penalization for the state equation for an elliptical optimal control problem

We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss–Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.     

Optimal control of constrained time lag systems: Necessary conditions and numerical treatment

We consider retarded optimal control problems with constant delays in state and control variables under mixed controlstate inequality constraints. First order necessary optimality conditions in the form of Pontryagin's minimum principle are presented and discussed as well as numerical methods based upon discretization techniques and nonlinear programming. The minimum principle for the considered problem class leads to a boundary value problem which is retarded in the state dynamics and advanced in the costate dynamics. It can be shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Necessary conditions for functional differential inclusions

We consider an optimal control problem in which the dynamic equation and cost function depend on the recent past of the trajectory. The regularity assumed in the basic data is Lipschitz continuity with respect to the sup norm. It is shown that, for a given optimal solution, an adjoint arc of bounded variation exists that satisfies an associated Hamiltonian inclusion. From this result, known smooth versions of the Pontryagin maximum principle for hereditary problems can be easily derived. Problems with Euclidean endpoint constraints are also considered.     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

   Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint.     

Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.