Mixed Control Problem Research Articles

A dam management problem with energy production as an optimal switching problem

AbstractWe consider an optimal stochastic control problem for a dam. Electrical power production is operating under an uncertain setting for electricity market prices and water level which has to be kept under control. Indeed, the water level inside the basin cannot exceed a certain threshold for safety reasons, and at the same time cannot decrease below another threshold in order to keep power production active. We model this situation as a mixed control problem with regular and switching controls under constraints. We characterize the value function as solution of an HJB equation and provide some numerical approximating methods. We shall illustrate by numerical examples the main achievements of the present approach.

Robust risk‐sensitive control

SummaryWe introduce arisk‐sensitivegeneralization of the mixed control problem for linear stochastic systems with additive noise. Two criteria ofexponential‐quadraticform are used to generalise the usual quadratic criteria. The solutions are found in a linear state‐feedback form for both the finite and the infinite horizon formulations in terms of coupled Riccati differential and algebraic equations. A change of measures for both criteria and completion of squares method is used to derive the solutions, and explicit sufficient conditions for the admissibility of controls are derived. An application to the problem ofrobustportfolio control in a market with random interest rate subject to a disturbance is also given.

Dual Control Methods for a Mixed Control Problem with Optimal Stopping Arising in Optimal Consumption and Investment

Dual Control Methods for a Mixed Control Problem with Optimal Stopping Arising in Optimal Consumption and Investment

Delta family approach for the stochastic control problems of utility maximization

In this paper, we propose a new approach for stochastic control problems arising from utility maximization. The main idea is to directly start from the dynamical programming equation and compute the conditional expectation using a novel representation of the conditional density function through the Dirac Delta function and the corresponding series representation. We obtain an explicit series representation of the value function, whose coefficients are expressed through integration of the value function at a later time point against a chosen basis function. Thus we are able to set up a recursive integration time-stepping scheme to compute the optimal value function given the known terminal condition, e.g. utility function. Due to tensor decomposition property of the Dirac Delta function in high dimensions, it is straightforward to extend our approach to solving high-dimensional stochastic control problems. The backward recursive nature of the method also allows for solving stochastic control and stopping problems, i.e. mixed control problems. We illustrate the method through solving some two-dimensional stochastic control (and stopping) problems, including the case under the classical and rough Heston stochastic volatility models, and stochastic local volatility models such as the stochastic alpha beta rho (SABR) model.

Stochastic recursive optimal control problem with obstacle constraint involving diffusion type control

This paper concerns a kind of stochastic optimal control problem with recursive utility described by a reflected backward stochastic differential equation (RBSDE, for short) involving diffusion type control which covers regular control problem, singular control problem and impulse control problem. To begin with, the existence and uniqueness of solution for RBSDEs involving diffusion type control is derived. Then, for the related recursive optimal control problem with obstacle constraint, a sufficient condition to obtain the optimal regular control and diffusion type control is provided. Hence, based on the connection between RBSDE and optimal stopping problem, a class of recursive optimal mixed control problem involving diffusion type control is considered to illustrate our theoretical result, and here the explicit optimal control as well as the stopping time are obtained.

Solution to Mixed H2/H∞ Control for Discrete–time Systems with (x,u,v)–dependent Noise

In this paper, the stochastic H2/H∞ control problem for linear discrete–time systems with (x,u,v)–dependent noise is studied. By applying the leader–follower stochastic game approach, the disturbance is treated as the follower and the control input is treated as the leader, respectively. Necessary and sufficient conditions for the mixed control problem are presented which guarantee the existence and uniqueness of the optimal solution. By applying the stochastic maximum principle, the follower first solves a stochastic linear quadratic optimal control problem which is given in the form of H∞–norm with the aid of stochastic Riccati equations. Then the leader solves a stochastic linear quadratic problem with the aid of forward and backward equations. The main technique is to introduce two new co–states to capture the future information, the encountered difficulty is to establish a homogeneous relationship between the new co–states.

Mixed Control Problem for the Linearized Quasi-Stationary Phase Field System of Equations

Conditions are obtained for unique solution existence of a mixed control problem without taking in account control expenses for a system that described by an initial-boundary value problem for the linearized quasi-stationary system of phase field equations. The problem is reduced to an optimal control problem for operator differential equation of first order in abstract space with degenerate operator at derivative using start and distributed controls simultaneously. The theorem on the unique solvability of this problem is applied to studying of optimal control problem for the phase field equations system.

Численный метод решения задач смешанного управления для систем леонтьевского типа

The article presents exact and approximate solutions of mixed control problems. A detailed algorithm for the numerical method of the solution of mixed control problems is presented, the convergence of approximate solutions to the exact one is proved. Methods of the theory of degenerate (semi) groups, of the optimal control theory are used. The importance of the introduced functional quality which enables to solve applied problems in economics and engineering is shown

The Existence of a Unique Solution to a Mixed Control Problem for Sobolev-Type Equations

This article studies a mixed control problem for Sobolev-type equations in the case of a relatively radial operator. We use the Showalter Sidorov initial condition. The di erence in the statement of our problem from those studied previously by other researchers amounts to the form of the quality functional, which, in the authors' opinion, is more adequate to model applications in economics and technology. We prove an existence and uniqueness theorem for the solution to this problem.

Computing Prager's Kinematic Hardening Mixed-Control Equations in a Pseudo-Riemann Manifold

Materials' internal spacetime may bear cer- tain similarities with the external spacetime of special relativity theory. Previously, it is shown that material hardening and anisotropy may cause the internal space- time curved. In this paper we announce the third mech- anism of mixed-control to cause the curvedness of inter- nal spacetime. To tackle the mixed-control problem for a Prager kinematic hardening material, we demonstrate two new formulations. By using two-integrating factors idea we can derive two Lie type systems in the product space of M m+1 ⊗M n+1 . The Lie algebra is a direct sum of so(m, 1) ⊕so(n, 1), and correspondingly the symme- try group is a direct product of SOo(m, 1) ⊗SOo(n, 1), which left acts on a twin-cone. Then, by using the one- integrating factor idea we can convert the nonlinear con- stitutive equations into a Lie type system of u Z=CZ with C ∈sl(5, 1,R) a Lie algebra of the special orthochronous pseudo-linear group SL(5, 1,R). The underlying space is a distorted cone in the pseudo-Riemann manifold. Con- sistent numerical methods are then developed according to these Lie symmetries, and numerical examples are used to assess the performance of new algorithms. The measures in terms of the errors by satisfying the consis- tency condition, strain and stress relative errors and ori- entational errors confirm that the new numerical methods are better than radial return method.

Mixed Optimal Stopping and Stochastic Control Problems with Semicontinuous Final Reward for Diffusion Processes

We consider mixed control problems for diffusion processes, i.e. problems which involve both optimal control and stopping. The running reward is assumed to be smooth, but the stopping reward need only be semicontinuous. We show that, under suitable conditions, the value function w has the same regularity as the final reward g, i.e. w is lower or upper semicontinuous if g is. Furthermore, when g is l.s.c., we prove that the value function is a viscosity solution of the associated variational inequality.

Estimation of Pareto sets in the mixed control problem

Multiobjective design problems give rise to a well-defined object: the Pareto-set. This paper proposes some verifiable conditions that are applicable to sets with finite number of elements, to corroborate or falsify the hypothesis of the elements of that set being samples of the Pareto set. These conditions lead to several generic criteria that can be employed in the evaluation of algorithms as multiobjective optimization mechanisms. A conceptual multiobjective genetic algorithm is proposed, exploiting the group properties of the intermediate Pareto-set estimates to generate a consistent final estimate. The methodology is applied to the case of a mixed control design. Recent dedicated multiobjective algorithms are evaluated under the proposed methodology, and it is shown that they can generate sub-optimal or non-consistent solution sets. It is shown that the proposed synthesis methodology can lead to both enhanced objectives and enhanced consistency in the Pareto-set estimate.

Reflected BSDE's with discontinuous barrier and application

We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems.

Robust and optimal fuzzy control: A linear matrix inequality approach

This paper presents a mixed control design of robust fuzzy control and optimal fuzzy control based on relaxed stability conditions represented by linear matrix inequalities (LMIs). A robust fuzzy controller is designed so as to maximize the norm of the uncertain blocks in a Takagi-Sugeno fuzzy model. Next, an optimal fuzzy controller is designed by solving the minimization problem that minimizes the upper bound of a given quadratic performance index. A mixed control problem that simultaneously considers both of them is defined and is efficiently solved via convex optimization techniques based on LMIs. Finally, a design example for a nonlinear control benchmark problem demonstrates the utility of the mixed control problem based on LMIs.

Active Vibration Control by Means of LMI-Based Mixed H2/H.INF. State Feedback Control.

In this paper, mixed H2 / H∞ control is studied using the linear matrix inequality approach(LMI). In many linear control problems, the design constraints can be reformulated in terms of LMI. In general, in H∞ control good frequency response performance is maintained but transient response of control system may not be guaranteed. Here, we have formulated H∞ and H2 control separately by means of LMI, and combined both objectives as a mixed control problem. We applied mixed H2 / H∞ control to an active vibration control problem and obtained reasonable results in frequency and time domains. The design approach described here is based on a numerical optimization technique which requires use of efficient convex optimization software.

Mixed control problem under partial observation

In this paper, we consider the control problem with optimal stopping of a jump process. Using compactification methods, we obtain the existence of an optimal Markovian optimal control.