OPTIMAL CONTROL OF MARKOVIAN SWITCHING SYSTEMS WITH APPLICATIONS TO PORTFOLIO DECISIONS UNDER INFLATION

This article is concerned with a class of control systems with Markovian switching,in which an ltd formula for Markov-modulated processes is derived.Moreover,an optimal control law satisfying the generalized Hamilton-Jacobi-Bellman(HJB) equation with Markovian switching is characterized.Then,through the generalized HJB equation,we study an optimal consumption and portfolio problem with the financial markets of Markovian switching and inflation.Thus,we deduce the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds.Finally,for the CRRA utility function,we explicitly give the optimal consumption and portfolio policies.Numerical examples are included to illustrate the obtained results.     

SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

Asymptotically optimal dividend policy for regime-switching compound Poisson models

This work develops asymptotically optimal dividend policies to maximize the expected present value of dividends until ruin.Compound Poisson processes with regime switching are used to model the surplus and the switching（a continuous-time controlled Markov chain） represents random environment and other economic conditions.Assuming the switching to be fast varying together with suitable conditions,it is shown that the system has a limit that is an average with respect to the invariant measure of a related Markov chain.Under simple conditions,the optimal policy of the limit dividend strategy is a threshold policy.Using the optimal policy of the limit system as a guide,feedback control for the original surplus is then developed.It is demonstrated that the constructed dividend policy is asymptotically optimal.     

A Stability Theorem for Constrained Optimal Control Problems

This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.     

Optimal Delaunay Triangulations

The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in L^P-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function ‖x‖^2 among all the triangulations with a given set of vertices. For a more general function, a functiondependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-depend entoptimal Delaunay triangulation is proved to exist for any given convex continuous function.On an optimal Delaunay triangulation associated with f, it is proved that △↓f at the interior vertices can be exactly recovered by the function values on its neighboring vertices.Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.     

模糊多目标半定规划

This paper first applies the fuzzy set theory to multi-objective semi-definite program-ming （MSDP）, and proposes the fuzzy multi-objective semi-definite programming （FMSDP） model whose optimal efficient solution is defined for the first time, too. By constructing a membership function, the FMSDP is translated to the MSDP. Then we prove that the optimal efficient solution of FMSDP is consistent with the efficient solution of MSDP and present the optimality condition about these programming. At last, we give an algorithm for FMSDP by introducing a new membership function and a series of transformation.     

STABILITY ANALYSIS OF THE SPLIT-STEP THETA METHOD FOR NONLINEAR REGIME-SWITCHING JUMP SYSTEMS

In this paper,we investigate the stability of the split-step theta(SST)method for a class of nonlinear regime-switching jump systems–neutral stochastic delay differential equations(NSDDEs)with Markov switching and jumps.As we know,there are few results on the stability of numerical solutions for NSDDEs with Markov switching and jumps.The purpose of this paper is to enrich conclusions in such respect.It first devotes to show that the trivial solution of the NSDDE with Markov switching and jumps is exponentially mean square stable and asymptotically mean square stable under some suitable conditions.If the drift coefficient also satisfies the linear growth condition,it then proves that the SST method applied to the NSDDE with Markov switching and jumps shares the same conclusions with the exact solution.Moreover,a numerical example is demonstrated to illustrate the obtained results.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Single machine stochastic JIT scheduling problem subject to machine breakdowns

In this paper we research the single machine stochastic JIT scheduling problem subject to the machine breakdowns for preemptive-resume and preemptive-repeat.The objective function of the problem is the sum of squared deviations of the job-expected completion times from the due date.For preemptive-resume,we show that the optimal sequence of the SSDE problem is V-shaped with respect to expected processing times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.We discuss the difference between the SSDE problem and the ESSD problem and show that the optimal solution of the SSDE problem is a good approximate optimal solution of the ESSD problem,and the optimal solution of the SSDE problem is an optimal solution of the ESSD problem under some conditions.For preemptive-repeat,the stochastic JIT scheduling problem has not been solved since the variances of the completion times cannot be computed.We replace the ESSD problem by the SSDE problem.We show that the optimal sequence of the SSDE problem is V-shaped with respect to the expected occupying times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.A new thought is advanced for the research of the preemptive-repeat stochastic JIT scheduling problem.     

Deterministic Time-inconsistent Optimal Control Problems - an Essentially Cooperative Approach

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N -person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.     

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.     

Systems governed by ordinary differential equations with continuous,switching and impulse controls

Optimal control problem for systems governed by ordinary differential equations with continuous, switching and impulse controls are studied. It is proved that the value function of the problem is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman system.     

Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

We study an optimal control problem for a hybrid system exhibiting several internal switching variables whose discrete evolutions are governed by some delayed thermostatic laws. By the dynamic programming technique we prove that the value function is the unique viscosity solution of a system of several Hamilton-Jacobi equations, suitably coupled. The method involves a contraction principle and some suitably adapted results for exit-time problems with discontinuous exit cost.     

Classical and Impulse Control for the Optimization of Dividend and Proportional Reinsurance Policies with Regime Switching

We consider the optimal proportional reinsurance and dividend strategy. The surplus process is modeled by the classical compound Poisson risk model with regime switching. Considering a class of utility functions, the object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted utility of the shareholders until ruin. By adapting the techniques and methods of stochastic control, we study the quasi-variational inequality for this classical and impulse control problem and establish a verification theorem. We show that the optimal value function is characterized as the unique viscosity solution of the corresponding quasi-variational inequality.     

Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model

In this article, we consider the optimal reinsurance and dividend strategy for an insurer. We model the surplus process of the insurer by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain. The object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted dividend payments until ruin. We give the definition of viscosity solution in the presence of regime switching. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and a verification theorem is also obtained.     

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.     

On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints

We study a quasi-variational inequality system with unbounded solutions. It represents the Bellman equation associated with an optimal switching control problem with state constraints arising from production engineering. We show that the optimal cost is the unique viscosity solution of the system.This work was supported by the National Research Council of Argentina, Grant No. PID-BID 213.     

Parameter Tuning of Multi-Proportional-Integral-Derivative Controllers Based on Optimal Switching Algorithms

Under the framework of switched systems, this paper considers a multi-proportional-integral-derivative controller parameter tuning problem with terminal equality constraints and continuous-time inequality constraints. The switching time and controller parameters are decision variables to be chosen optimally. Firstly, we transform the optimal control problem into an equivalent problem with fixed switching instants by introducing an auxiliary function and a time-scaling transformation. Because of the complexity of constraints, it is difficult to solve the problem by conventional optimization techniques. To overcome this difficulty, a novel exact penalty function is introduced for these constraints. Furthermore, the penalty function is appended to the cost functional to form an augmented cost functional, giving rise to an approximate nonlinear parameter optimization problem that can be solved using any gradient-based method. Convergence results indicate that any local optimal solution of the approximate problem is also a local optimal solution of the original problem as long as the penalty parameter is sufficiently large. Finally, an example is provided to illustrate the effectiveness of the developed algorithm.     

A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit Runge–Kutta integration

Efficient and reliable integrators are indispensable for the design of sequential solvers for optimal control problems involving continuous dynamics, especially for real-time applications. In this paper, optimal control problems for systems represented by index-1 differential-algebraic equations are investigated. On the basis of a time-scaling transformation, the control is parameterized as a piecewise constant function with variable heights and switching time instants. Compared with control parameterization with fixed time grids, the flexibility of adjusting switching time instants increases the chance of finding the optimal solution. Furthermore, error constraints are introduced in the optimization procedure such that the optimal control obtained has a guarantee of integration accuracy. For the derived approximate nonlinear programming problem, a function evaluation and forward sensitivity propagation algorithm is proposed with an embedded implicit Runge–Kutta integrator, which executes one Newton iteration in the limit by employing a predictor-corrector strategy. This algorithm is combined with a nonlinear programming solver Ipopt to construct the optimal control solver. Numerical experiments for the solution of the optimal control problem for a Delta robot demonstrate that the computational speed of this solver is increased by a factor of 0.5–2 when compared with the same solver without the predictor-corrector strategy, and increased by a factor of 20–40 when compared with solver embedding IDAS, the Implicit Differential-Algebraic solver with Sensitivity capabilities developed by Lawrence Livermore National Laboratory. Meanwhile, the accuracy loss compared with the one using IDAS is small and admissible.     

半线性热方程的反馈零能控性

考虑具有Lipschitz非线性项,半线性热方程的最优控制问题.我们将运用观测不等式,证明值函数ψ作为相应Hamilton-Jacobi方程的唯一粘性正解是局部Lipschitz连续的.最后,运用动态规划方法,得到系统最优的反馈控制.