Optimality Conditions For Control Problems Research Articles

Optimality Conditions in Control Problems with Random State Constraints in Probabilistic or Almost Sure Form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution. Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].

New Equilibria and Dynamic Structures Under Continuous Optimal Feedback Control

In this paper, the new equilibria realized by continuous optimal control inputs and the dynamic structure around them are studied. Using the Euler–Lagrange equation, which is a necessary condition for optimal control problems, the equations of motion of a dynamic system with optimal control inputs that minimize the quadratic cost function are described in terms of state and adjoint variables. Based on the equations of motion, equilibrium conditions are derived, and the properties of equilibria are analyzed for the two-body and Hill three-body problems. The stability and dynamic structure around unstable equilibria are also characterized to get insights into the properties of optimal trajectories.

Network optimality conditions

Abstract Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models. Therefore, we consider only control systems described by ordinary differential equations. First, we derive optimality conditions for the nonlinear problem for a single beam. These conditions are formulated in terms of the local Pontryagin maximum principle and the matrix Riccati equation. Then, the optimality conditions for the control problem for networks posed on an arbitrary planar graph are discussed. This problem has a set of independent variables x i varying within their intervals [0, l i], associated with the corresponding beams at network edges. The lengths l i of intervals are not specified and must be determined. So, the optimization problem is non-standard, it is a combination of control and design of networks. However, using a linear change of the independent variables, it can be reduced to a standard one, and we show this. Two simple numerical examples for the single-beam problem are considered.

Analytical approaches for growth models in economics

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed-form) solutions and graphical representations for the optimal control problems are represented.

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

AbstractWe study the stochastic optimal control problem for fully coupled forward‐backward stochastic differential equations (FBSDEs) with jump diffusions. A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the backward SDE and the presence of jump diffusions. Previously, this class of problems has been solved via only the stochastic maximum principle, which guarantees only the necessary condition of optimality and requires identifying unknown parameters in the corresponding variational inequality. Our paper provides an alternative approach, which constitutes the sufficient condition for optimality. Specifically, the original fully coupled FBSDE control problem (referred to as (P)) is converted into the terminal state‐constrained forward stochastic control problem (referred to as ) that includes additional (possibly unbounded) control variables. Then is solved via the backward reachability analysis, by which the value function of is expressed as the zero‐level set of the value function for the auxiliary unconstrained (forward) control problem (referred to as ). Unlike ), is an unconstrained problem, which includes additional control variables as a consequence of the martingale representation theorem. We show that the value function for is the unique viscosity solution to the associated integro‐type Hamilton‐Jacobi‐Bellman (HJB) equation. The viscosity solution analysis presented in our paper requires a new technique due to additional control variables in the Hamiltonian maximization and the presence of the nonlocal integral operator in terms of the (singular) Lévy measure. To solve the original problem (P), we reverse our approach. Specifically, we first solve to obtain the value function using the verification theorem and the viscosity solution of the HJB equation. Then is solved by characterizing the zero‐level set of the value function of , from which the optimal solution of (P) can be constructed. To illustrate the theoretical results of this paper, applications to the linear‐quadratic problem for fully coupled FBSDEs with jumps are also presented.

Second-Order Necessary Conditions for Optimal Control Problems with Endpoints-Constraints and Convex Control-Constraints

Second-Order Necessary Conditions for Optimal Control Problems with Endpoints-Constraints and Convex Control-Constraints

Optimal control analysis of a multigroup SEAIHRD model for COVID-19 epidemic.

The COVID‐19 pandemic has threatened public health and caused substantial economic loss to most countries worldwide. A multigroup susceptible–exposed–asymptomatic–infectious–hospitalized–recovered–dead (SEAIHRD) compartment model is first constructed to model the spread of the disease by dividing the population into three age groups: young (aged 0–19), prime (aged 20–64), and elderly (aged 65 and over). Then, we develop a free terminal time, partially fixed terminal state optimal control problem to minimize deaths and costs associated with hospitalization and the implementation of different control strategies. And the optimal strategies are derived under different assumptions about medical resources and vaccination. Specifically, we explore optimal control strategies for reaching herd immunity in the COVID‐19 outbreak in a free terminal time situation to evaluate the effect of nonpharmaceutical interventions (NPIs) and vaccination as control measures. The transmission rate of SARS‐CoV‐2 is calibrated by using real data in the United States at the early stage of the epidemic. Through numerical simulation, we conclude that the outbreak of COVID‐19 can be contained by implementing appropriate control of the prime age population and relatively strict control measures for young and elderly populations. Within a specific period, strict control measures should be implemented before the vaccine is marketed.

State and Control Path-Dependent Stochastic Optimal Control With Jumps

We consider the state and control path-dependent stochastic optimal control problem for jump-diffusion models, where the dynamics and the objective functional are dependent on (current and past) paths of state and control processes. We prove the dynamic programming principle of the value functional, for which, unlike the existing literature, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Skorohod metric</i> is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the state and control path-dependent integro-type Hamilton–Jacobi–Bellman (PIHJB) equation, which includes the Lévy measure in the corresponding nonlocal path-dependent integral operator. Then, by using the functional Itô calculus of a càdlàg path, we show the verification theorem, which constitutes the sufficient condition for optimality in terms of the solution to the PIHJB equation. We finally apply our verification theorem to the linear-quadratic optimal control problem of jump-diffusion models with delay and the control path-dependent problem, for which the explicit optimal solutions are obtained by solving the corresponding PIHJB equation.

Linear Pseudospectral Method with Chebyshev Collocation for Optimal Control Problems with Unspecified Terminal Time

In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application.

A global convergent semi-smooth Newton method for semi-linear elliptic optimal control problem

Global convergent semi-smooth Newton (GCSSN) method for L 2 -norm control constrained elliptic optimal control problem with L 1 -control cost is discussed. The first order necessary optimality conditions are analyzed and reduced system is proposed. After finite difference discretization, we propose the global convergent semi-smooth Newton method for discrete reduced system with the nonmonotone line search. The local superlinear convergence rate and convergence are proved theoretically. The demonstrated theoretical properties are verified with numerical results. To illustrate the effectiveness and efficiency, we compare the proposed method with semi-smooth Newton method in the simulation part. • Global convergent semi-smooth Newton method for semi-linear elliptic optimal control problem is proposed. • We obtain the first order optimality necessary condition for optimal control problem. • The convergence properties of algorithm are proven theoretically and verified by simulations.

Necessary conditions for optimal control problems with sweeping systems and end point constraints

We generalize the Maximum Principle for free end point optimal control problems involving sweeping systems derived in [de Pinho MdR, Ferreira MMA, Smirnov GV. Optimal control involving sweeping processes. Set-Valued Var Anal. 2019;27(2):523–548] to cover the case where the constraints are time dependent and the end point is constrained to a set. As in [de Pinho MdR, Ferreira MMA, Smirnov GV. Optimal control involving sweeping processes. Set-Valued Var Anal. 2019;27(2):523–548], an ingenious smooth approximating family of standard differential equations plays a crucial role.

Subdifferentiation of Nonconvex Sparsity-Promoting Functionals on Lebesgue Spaces

Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that optimal controls vanish on large parts of the underlying domain. Typical candidates for those terms are integral functions on Lebesgue spaces based on the $\ell_p$-metric for $p\in[0,1)$ which are nonconvex as well as non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas for the Frechet, limiting, and singular subdifferential of these functionals. These generalized derivatives can be used for the derivation of necessary optimality conditions for optimal control problems comprising such sparsity-promoting terms.

Constrained optimization problems governed by PDE models of grain boundary motions

Abstract In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.

Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints

This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an ε-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained.

Some Remarks on the Issue of Second-order Optimality Conditions in Control Problems with Mixed Constraints

A smooth optimal control problem with mixed constraints is considered. Under the normality assumption, a proof of second-order necessary optimality conditions based on the Robinson stability theorem is provided. The main feature of the obtained result is that the local regularity with respect to the mixed constraints is assumed, that is, a regularity in an ε-tube along the minimizer, but not the conventional global regularity hypothesis. This impacts the maximum condition. Therefore, the normal set of Lagrange multipliers fulfills the Legendre-Clebsch condition and the maximum principle. At the same time, the maximum condition is modified since, now, the maximum is taken over a reduced feasible set. Furthermore, the case of abnormal minimizers is considered. The same type of reduced maximum condition is obtained along with a refined Legendre-Clebsch condition which is meaningful in the abnormal case.

First order necessary condition for stochastic evolution control systems with random generators

&lt;p style='text-indent:20px;'&gt;The main purpose of this paper is to establish the first order necessary optimality condition for optimal control problems of stochastic evolution systems with random generators. For our controlled system, both the drift and the diffusion terms contain the control variable, and the control region is a nonempty closed subset of a separable Hilbert space. Compared with the existing works, the main difficulties here are to prove the well-posedness of the control and the adjoint systems. We obtain the well-posedness of the adjoint system in the sense of mild solutions and that of the control system by means of the stochastic transposition method. In addition, the variational analysis approach is employed to handle the nonconvexity of the control region when deriving our optimality condition.&lt;/p&gt;

Correction to: Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

Correction to: Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

Deep Learning for Efficient and Optimal Motion Planning for AUVs with Disturbances.

We use the recent advances in Deep Learning to solve an underwater motion planning problem by making use of optimal control tools—namely, we propose using the Deep Galerkin Method (DGM) to approximate the Hamilton–Jacobi–Bellman PDE that can be used to solve continuous time and state optimal control problems. In order to make our approach more realistic, we consider that there are disturbances in the underwater medium that affect the trajectory of the autonomous vehicle. After adapting DGM by making use of a surrogate approach, our results show that our method is able to efficiently solve the proposed problem, providing large improvements over a baseline control in terms of costs, especially in the case in which the disturbances effects are more significant.

ALGORITHMS AND SOFTWARE OF BUILDINGS INTERMITTENT THERMAL CONDITIONS OPTIMAL CONTROL

The actual problem of synthesizing the optimal control law for the intermittent heating mode of office buildings is considered, which consists in the programmed reduction of the room temperature during off-hours. The analysis of information sources showed the absence of sufficiently justified methods for solving the problem of optimizing the thermal processes of buildings, which provide comfortable conditions for the functioning of personnel, on the one hand, and save costs for buildings maintenance, on the other hand. The statement of the building thermal state optimal control problem is formulated, which contains a mathematical model of the process, a control goal, constraints on control actions and an integral quadratic quality criterion, including deviations of the room temperature from a given program and heat consumption. The limitation on the control action is taken into account by the method of penalty functions. As a mathematical model of the optimized thermal process, a two-dimensional controlled system of differential equations with constant coefficients is applied. The values of these coefficients are determined on the basis of building similarity criteria, which allows solving the optimization problem in a dimensionless form. The model includes a control action - the supplied heat flux, and a disturbing action - the ambient temperature, changing according to a given harmonious law. The numerical solution of the optimal control problem is implemented by the Chernous'ko method of successive approximations. A description of the user interface is given, as well as examples of the implementation of the thermal state optimal control. The proposed software allows on the stage of designing the heating system to obtain express assessments of various thermal modes of the building, and can also be used as computer support for automated individual heating points.&#x0D; Keywords: intermittent heating mode, optimal control, mathematical model, control quality criterion, control constraints, penalty function, successive approximations, program interface, automated individual heating point.

Second order optimality conditions for periodic optimal control problems governed by semilinear parabolic differential equations

In this paper, we study second-order optimality conditions for some optimal control problems governed by some semi-linear parabolic equations with periodic state constraint in time. We obtain a necessary condition and a sufficient condition in terms of the second order derivative of the associated Lagrangian. These two conditions correspond to the positive definite and the nonnegativity of the second order derivative of the Lagrangian on the same cone, respectively.