Control-state Constraints Research Articles

Constrained Utility Deviation-Risk Optimization and Time-Consistent HJB Equation

In this paper we propose a unified utility deviation-risk model which covers both utility maximization and mean-variance analysis as special cases. We derive the time-consistent Hamilton--Jacobi--Bellman (HJB) equation for the equilibrium value function and significantly reduce the number of state variables, which makes the HJB equation derived in this paper much easier to solve than the extended HJB equation in the literature. We illustrate the usefulness of the time-consistent HJB equation with several examples which recover the known results in the literature and go beyond, including a mean-variance model with stochastic volatility dependent risk aversion, a utility deviation-risk model with state dependent risk aversion and control constraint, and a constrained portfolio selection model. The numerical and statistical tests show that the utility and deviation-risk have a significant impact on the equilibrium control strategy and the distribution of the terminal wealth.

Convergence Analysis for Approximations of Optimal Control Problems Subject to Higher Index Differential-Algebraic Equations and Mixed Control-State Constraints

This paper establishes a convergence result for implicit Euler discretizations of optimal control problems with differential-algebraic equations of higher index and mixed control-state constraints. The main difficulty of the analysis is caused by a structural discrepancy between the necessary conditions of the continuous optimal control problem and the necessary conditions of the discretized problems. This discrepancy does not allow one to compare the respective necessary conditions directly. We use an equivalent reformulation of the discretized problems to overcome this discrepancy and to prove first order convergence of the discretized states, algebraic states, controls, and multipliers of the reformulation.

Sparse optimal control for the heat equation with mixed control-state constraints

A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the \begin{document}$ L^1 $\end{document} -norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.

Necessary Second-Order Optimality Conditions in Distributed-Boundary Semilinear Elliptic Optimal Control with Twice Directionally Differentiable Functions

In this paper we investigate second-order optimality conditions for distributed-boundary control problems governed by semilinear elliptic systems, where the objective function may not be of class \(C^{2}\), in the presence of pointwise pure state constraints and pointwise mixed state-control constraints. Using local approximations and directional derivatives for twice directionally differentiable mappings, we examine necessary second-order optimality conditions for the problems, by virtue of a Lagrange multiplier theorem in mathematical programs.

Open-loop and feedback Nash equilibria in constrained linear–quadratic dynamic games played over event trees

We characterize open-loop and feedback Nash equilibria for a class of constrained linear–quadratic multistage games having the following features: (a) The control variables are of two types, namely, control variables that enter the dynamics, but are not constrained, and control variables that are part of state-control constraints, but do not enter the dynamics; (b) The parameter values are uncertain, with the stochastic process being described by a event tree.This paper takes stock on Reddy and Zaccour (2015, 2017) where the same class of games was considered, but in a deterministic setting. Here, we follow the same approaches and rewrite the results in these two papers in a stochastic setting.

Adaptive finite element methods for sparse PDE-constrained optimization

Abstract We propose and analyse reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method, whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform.

Pareto Fronts of the Set of Sustainable Thresholds for Constrained Control Systems

This paper is concerned with a dual object of the so-called viability kernel of a control system with mixed (state-control) constraints. This object, called the set of sustainable thresholds, describes the possible thresholds for which a given initial condition is viable. In this work, we are concerned with characterizing the weak and strong Pareto fronts of the set of sustainable thresholds and providing a practical method for computing such objects based on optimal control theory and a level-set approach. A numerical example, relying on renewable resource management, is shown to demonstrate the proposed method.

Second-order Necessary Optimality Conditions for an Optimal Control Problem

Second-order necessary optimality conditions for an optimal control problem with a nonconvex cost function and state-control constraints are studied in this paper. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we obtain second-order necessary optimality conditions for an optimal control problem.

Optimal Control of Endoatmospheric Launch Vehicle Systems: Geometric and Computational Issues

In this paper, we develop a geometric analysis and a numerical algorithm, based on indirect methods, to solve optimal guidance of endoatmospheric launch vehicle systems under mixed control-state constraints. Two main difficulties are addressed. First, we tackle the presence of Euler singularities by introducing a representation of the configuration manifold in appropriate local charts. In these local coordinates, not only the problem is free from Euler singularities but also it can be recast as an optimal control problem with only pure control constraints. The second issue concerns the initialization of the shooting method. We introduce a strategy that combines indirect methods with homotopies, thus providing high accuracy. We illustrate the efficiency of our approach by numerical simulations on missile interception problems under challenging scenarios.

On Lifting Operators and Regularity of Nonsmooth Newton Methods for Optimal Control Problems of Differential Algebraic Equations

This paper focuses on nonsmooth Newton methods of optimal control problems governed by mixed control–state constraints with differential algebraic equations. In contrast to previous results, we analyze lifting operators involved in nonsmooth Newton methods and establish corresponding convergence results. We also give sufficient conditions for regularity of generalized derivatives of systems of nonsmooth operator equations associated with optimal control problems.

Barrier Lyapunov functions-based localized adaptive neural control for nonlinear systems with state and asymmetric control constraints

A barrier Lyapunov functions (BLFs)-based localized adaptive neural network (NN) control is proposed for a class of uncertain nonlinear systems with state and asymmetric control constraints to track the reference trajectory. To handle system constraints, BLFs are used in the backstepping procedure, and the control input is considered as an extended state variable. This extends current research on BLFs-based control for systems with state and output constraints to systems with state and asymmetric control constraints. A locally weighted learning NN with projection modification is designed to estimate and compensate for the system uncertainty. The use of projection modification ensures the NN estimator is contained in a given bounded area and prevents the absolute value of NN output from being near to or larger than the bound of the tracking error. The feasibility and effectiveness of the proposed control have been demonstrated by formal proof and simulation results.

Sequence-based necessary second-order optimality conditions for semilinear elliptic optimal control problems with nonsmooth data

The main goal of this note is to formulate sequence-based necessary second-order optimality conditions for a semilinear elliptic optimal control problem, with a pointwise pure state constraint and a pointwise mixed state-control constraint, governed by nonsmooth functions. Using a modification of the Dubovitskii–Milyutin scheme, we develop necessary second-order conditions for optimality, which depend on given sequences, in terms of sequential tangent sets and sequential directional derivatives for not necessarily twice differentiable mappings.

Optimal vaccination strategy of a constrained time-varying SEIR epidemic model

The optimal control strategy for time-varying epidemic models remains a wide open research area. In this paper, the optimal vaccination strategy for a constrained time-varying SEIR (Susceptible, Exposed, Infected and Recovered) epidemic model is solved under the frame of constrained optimal control problems. Three time-varying factors, i.e., seasonally varying incidence coefficient, monotonic decreasing successfully immune rate and monotonic increasing vaccine yield are considered. Constraints on vaccination rate, susceptible population and vaccine supply at each time instant, which are pure-control constraint, pure-state constraint and mixed state-control constraint, respectively, are all taken into consideration. The characterization for the optimal control is derived with the help of the Pontryagin's maximum principle. And optimal control problems are successfully solved by a symplectic pseudospectral method numerically. Numerical results are consistent with the analytical ones. Finally, comparisons of different cases demonstrate that time-varying factors could alter the optimal vaccination strategy, which implies that omitting the time-varying factors may result in less optimal even unreasonable control strategy.

Research on the Maximum Range of Gliding Vehicles Based on Gauss Pseudo-spectral Method

Trajectory optimization is an important problem in the field of aircraft design. Effective trajectory optimization is critical to the tactical flight of an aircraft. Based on previous research, this paper further studies trajectory optimization of an unpowered gliding vehicle. The goal is to maximize its range and to analyze the influence of different initial conditions on trajectory optimization. In this paper, the trajectory optimization model involving all kinds of state constraints and control constraints was established and aerodynamic coefficient was calculated by CFD software. Finally Gauss Pseudo-spectral Method program was run to calculate the flight path of the gliding vehicle and to seek trajectory with maximum range. The simulation data shows that results of Gauss Pseudo-spectral Method are realistic and prove its practical operability. The use of Gauss Pseudo-spectral Method for trajectory optimization has a good effect.

The neuro-dynamic scheme for solving general form of discrete time optimal control problems

In this paper, we show that recently developed neural network methods for quadratic programming can be put to use in solving discrete time optimal control problems, with general pointwise constraints on states and controls. We describe a high performance recurrent neural network for a discrete time linear quadratic regulator problem with mixed state–control constraints. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the discrete time problem. It is also shown that the proposed network model is stable in the Lyapunov sense and it is globally convergent to an exact optimal solution of the original problem. Several practical examples are provided to show the feasibility and the efficiency of the scheme.

Optimal control problems with time delays: Two case studies in biomedicine.

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming L¹-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For L¹-type objectives we obtain extremal controls of bang-bang type.

Modeling and optimal control of HIV/AIDS prevention through PrEP

Pre-exposure prophylaxis (PrEP) consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals and has recently demonstrated to be highly efficacious for HIV prevention. We propose a new epidemiological model for HIV/AIDS transmission including PrEP. Existence, uniqueness and global stability of the disease free and endemic equilibriums are proved. The model with no PrEP is calibrated with the cumulative cases of infection by HIV and AIDS reported in Cape Verde from 1987 to 2014, showing that it predicts well such reality. An optimal control problem with a mixed state control constraint is then proposed and analyzed, where the control function represents the PrEP strategy and the mixed constraint models the fact that, due to PrEP costs, epidemic context and program coverage, the number of individuals under PrEP is limited at each instant of time. The objective is to determine the PrEP strategy that satisfies the mixed state control constraint and minimizes the number of individuals with pre-AIDS HIV-infection as well as the costs associated with PrEP. The optimal control problem is studied analytically. Through numerical simulations, we demonstrate that PrEP reduces HIV transmission significantly.

Optimal Control Problems with Terminal Complementarity Constraints

In this paper, we study an optimal control problem of ordinary differential equations with linear dynamics, affine mixed control-state constraints, and terminal complementarity constraints on the state function. We derive its weak, Mordukhovich, and strong stationarity conditions, and we present constraint qualifications which ensure that these conditions are satisfied at a locally optimal solution of the optimal control problem.

Metrically Regular Differential Generalized Equations

In this paper we consider a control system coupled with a generalized equation, which we call a differential generalized equation (DGE). This model covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We study metric regularity and strong metric regularity of mappings associated with DGE by focusing in particular on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts. Metric regularity of a control system subject to inequality state-control constraints is characterized. A sufficient condition for local controllability of a nonlinear system is obtained via metric regularity. Sufficient conditions for strong metric regularity in function spaces are presented in terms of uniform pointwise strong metric regularity. A characterization of the Lipschitz continuity of the control part of the solution mapping as a function of time is established. Finally, a path-following procedure for a discretized DGE is proposed for which an error estimate is derived.

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

ABSTRACTAn optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.