Optimal dividend payments in the stochastic Ramsey model

We consider the dividend payments of a self-financing firm in the stochastic Ramsey model. The firm invests in capital stock and its production technology is given by the Cobb–Douglas function. Our objective is to maximize the expected present value of future real dividends subject to a positive constraint on the capital stock. We use the penalization method to obtain a solution for the variational inequality associated with the optimal growth problem and give a synthesis of the optimal dividend policy.     

Optimal control in a mathematical model for leukemia therapy with phase constraints

We examine the optimal control problem that arises in the mathematical modeling of leukemia therapy, to solve which the Pontryagin maximum principle and the penalty function method are employed. It is assumed that the drug is capable of killing not only diseased cells, but healthy cells as well. The character of the drug??s interaction with cells is described by appropriate therapy functions.     

Optimal trajectories of infinite-horizon deterministic control systems

We study the infinite-horizon deterministic control problem of minimizing ₀ ^T L(z, ) dt, T, whereL(z, ·) is convex in for fixedz but not necessarily jointly convex in (z, ). We prove the existence of a solution to the infinite-horizon Bellman equation and use it to define a differential inclusion, which reduces in certain cases to an ordinary differential equation. We discuss cases where solutions of this differential inclusion (equation) provide optimal solutions (in the overtaking optimality sense) to the optimization problem.A quantity of special interest is the minimal long-run average-cost growth rate. We compute it explicitly and show that it is equal to min _x L(x, 0) in the following two cases: one is the scalar casen = 1 and the other is' when the integrand is in a separated form      

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if b is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined ε within the time horizon T. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if $b$ is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined $\epsilon$ within the time horizon $T$. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

Optimal control by random sequences with constraints

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 143–145, June, 1991.     

Optimal portfolio execution under time-varying liquidity constraints

In this article, we take an algorithmic approach to solve the problem of optimal execution under time-varying constraints on the depth of a limit order book (LOB). Our algorithms are within the resilience model proposed by Obizhaeva and Wang (2013) with a more realistic assumption on the order book depth; the amount of liquidity provided by an LOB market is finite at all times. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a one-dimensional root-finding problem which can be readily solved by standard numerical algorithms. When the depth of the order book is monotone in time, we apply the Karush-Kuhn-Tucker conditions to narrow down the set of candidate strategies. Then, we use a dichotomy-based search algorithm to pin down the optimal one. For the general case, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.     

Optimal control over the heat field in thin bodies local constraints on control

Asymptotic solutions of problems of optimal locally constrained control over the heat field in thin bodies are constructed and justified. These problems relate to the critical case of singularly perturbed systems (the degenerate problem has a family of solutions). Dnepropetrovsk Technical University of Railway Transport, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1200–1208, September, 1996.     

Optimal control of a big financial company with debt liability under bankrupt probability constraints

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company’s bankruptcy soon. In order to protect the shareholders’ profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, partial differential equation and variational inequality approach. Moreover, we establish a riskbased capital standard to ensure the capital requirement can cover the total given risk by numerical analysis, and give reasonable economic interpretation for the results.     

Optimal gathering protocols on paths under interference constraints

We study the problem of gathering information from the nodes of a multi-hop radio network into a predefined destination node under reachability and interference constraints. In such a network, a node is able to send messages to other nodes within reception distance, but doing so it might create interference with other communications. Thus, a message can only be properly received if the receiver is reachable from the sender and there is no interference from another message being transmitted simultaneously. The network is modeled as a graph, where the vertices represent the nodes of the network and the edges, the possible communications. The interference constraint is modeled by a fixed integer d≥1, which implies that nodes within distance d in the graph from one sender cannot receive messages from another node. In this paper, we suppose that each node has one unit-length message to transmit and, furthermore, we suppose that it takes one unit of time (slot) to transmit a unit-length message and during such a slot we can have only calls which do not interfere (called compatible calls). A set of compatible calls is referred to as a round. We give protocols and lower bounds on the minimum number of rounds for the gathering problem when the network is a path and the destination node is either at one end or at the center of the path. These protocols are shown to be optimal for any d in the first case, and for 1≤d≤4, in the second case.     

Non-compact-valued stochastic control under state constraints

In the present paper, we study a necessary condition under which the solutions of a stochastic differential equation governed by unbounded control processes, remain in an arbitrarily small neighborhood of a given set of constraints. We prove that, in comparison to the classical constrained control problem with bounded control processes, a further assumption on the growth of control processes is needed in order to obtain a necessary and sufficient condition in terms of viscosity solution of the associated Hamilton-Jacobi-Bellman equation. A rather general example illustrates our main result.     

Optimal control in coefficients of elliptic equations with state constraints

This paper is concerned with state constrained optimal control problems of elliptic equations, the control being a coefficient of the partial differential equation. Existence of an optimal control is proved and optimality conditions are derived. We perform finite-element approximations of optimal control problems and state some convergence results: we prove convergence of optimal controls and states as well as convergence of Lagrange multipliers.This research was partially supported by the Dirección General de Investigación Científica y Técnica (Madrid).     

Optimal synthesis in a two-dimensional problem with asymmetric constraints on the control

A two-dimensional optimal control problem is considered on the assumption that the terminal time of the process is not fixed and the integral objective functional depends on a parameter. Asymmetric constraints are imposed on the control parameter. Two cases are considered: constraints of the same sign and constraints of different signs. In the case of constraints of different signs, if the parameters of the problem satisfy certain relations, one obtains chattering control, alternating with a control with two switchings and a first-order singular are when these relations are violated. In the case of sign-definite control the controllability domain is part of the plane bounded by two semiparabolas. Three types of control law are then possible, in two of which the system will hit the boundary of the controllability domain and move along it, while the third features a first-order singular are. As the parameter of the problem is varied, the phase portrait undergoes evolution and one of these three types is interchanged with another. The optimality of these control laws is rigorously established using a dynamic programming method.     

Optimal control of damped Klein-Gordon equations with state constraints

In this paper, we study the maximum principles for optimal control problems governed by the damped Klein-Gordon equations with state constraints. And we prove the existence of the optimal parameter and deduce the necessary conditions on the optimal parameter.     

Optimal impulsive control problems with state and mixed constraints

Doklady Mathematics -     

The closure of the sheaf of trajectories of a linear control system with integral constraints

We consider a linear system with discontinuous coefficients controlled by a parameter under an integral constraint imposed on the control resource. It is well known that in such problems the closure of the sheaf of trajectories that correspond to ordinary controls (piecewise constant or measurable functions) coincides with the sheaf of trajectories in a generalized problem, where for generalized controls one uses finite additive measures of bounded variation. Therewith the closure is defined in the topology of pointwise convergence, because the limit elements (the generalized trajectories) may be discontinuous functions. In this paper we prove that any generalized trajectory can be approximated by a sequence of ordinary solutions to the initial system. We propose a concrete technique for constructing such sequences.