Maximum Principle For Optimal Control Research Articles

The Global Maximum Principle for Optimal Control of Partially Observed Stochastic Systems Driven by Fractional Brownian Motion

The Global Maximum Principle for Optimal Control of Partially Observed Stochastic Systems Driven by Fractional Brownian Motion

Maximum Principle for Optimal Control of Stochastic Evolution Equations with Recursive Utilities

Maximum Principle for Optimal Control of Stochastic Evolution Equations with Recursive Utilities

Integrable Systems: In the Footprints of the Greats

In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal.

Rolling Geodesics, Mechanical Systems and Elastic Curves

This paper defines a large class of differentiable manifolds that house two distinct optimal problems called affine-quadratic and rolling problem. We show remarkable connections between these two problems manifested by the associated Hamiltonians obtained by the Maximum Principle of optimal control. We also show that each of these Hamiltonians is completely intergrable, in the sense of Liouville. Finally we demonstrate the significance of these results for the theory of mechanical systems.

Maximum Principle for Optimal Control of COVID-19 Spread

Covid-19 is a disease caused by a new corona virus that has spread throughout the world and become a pandemic. It is classified as an infectious disease that can be transmitted from human to human through droplets. So that we need controls to reduce the spread of Covid-19. The optimal control that will be carried out this work is self-precaution, treatment and quarantined that will be applied to the dynamical modelling of Covid-19 spread using the Pontryagin’s Maximum Principle (PMP) to find out the optimal solution for the control. According to this principle the optimal control, corresponding optimal state, and adjoint function must minimize the Hamiltonian function. PMP converts the optimal control problem into a multipoint boundary value problem. That is, the optimality condition results in control. The optimal control variable, corresponding state and adjoint can be computed by solving an Ordinary Differential Equation system. The control strategies is aimed to reduce covid-19 transmission. Numerical results show the effectiveness of the control strategies in reducing Covid-19 spread. It is found that self-precaution more effective than treatment and quarantined.

A general maximum principle for progressive optimal stochastic control problems with Markov regime-switching

In this paper, we give a general maximum principle for optimal controls of stochastic systems driven by Markov chains. The control is allowed to enter both diffusion and jump terms and the control domain is not necessarily convex. We apply a new spike variation and the stochastic integral of progressive processes to obtain the main result.

The Maximum Principle for Optimal Control of BSDEs with Locally Lipschitz Coefficients

The Maximum Principle for Optimal Control of BSDEs with Locally Lipschitz Coefficients

Pontryagin's maximum principle for optimal control of an extrusion process

In this article, two optimal control problems of an extrusion process are considered in the isothermal case. The controlled system consists of a hyperbolic partial differential equation coupled with a nonlinear ordinary differential equation. Equipped with two control variables, the system describes the evolution of a moving interface between a fully filled zone and a partially filled zone. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of the Pontryagin maximum principle for the optimal control problem. In both fixed and free final horizon cases, the corresponding necessary optimality conditions are, respectively, established. A remark is then made for illustrating the applicability of the obtained results.

Maximum principle for optimal control of SPDEs with locally monotone coefficients

ABSTRACT The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we adapt the method which uses the relation between backward stochastic partial differential equation (BSPDE) and the maximum principle. In particular, necessary conditions for optimality for this stochastic optimal control problem are obtained. In spite of the fact that the method used here was used by several authors before, our adaptation is not immediate. It applies a trick which is used to get estimates for solutions of SPDE with Locally Monotone Coefficients as in the proof of the Lemmas 5.1 and 5.3. This adaptation permits us to apply our results to get a maximum principle for the optimal control to the cases when the system is governed by the 2D stochastic Navier-Stokes equation and by a stochastic Burgers' equation.

Maximum principle of optimal control of a Cahn–Hilliard–Navier–Stokes model with state constraints

AbstractWe investigate in this article Pontryagin's maximum principle for a class of control problems associated with a Cahn–Hilliard–Navier–Stokes model in a two dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity v, coupled with a Cahn–Hilliard model for the order (phase) parameter . We derive Pontryagin's maximum principle for the control problems assuming that a solution exists. Let us note that the coupling between the Navier–Stokes and the Cahn–Hilliard systems makes the analysis of the control problem more involved.

Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations

This work concerns some optimal control problems associated with the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn–Hilliard–Navier–Stokes model consists of a Navier–Stokes equation governing the fluid velocity field coupled with a convective Cahn–Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another optimal control problem of finding the unknown optimal initial data. Optimal data initialization problem is also known as the data assimilation problems in meteorology, where determining the correct initial condition is very crucial for the future predictions.

A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

In this paper we prove a maximum principle of optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps in the case where the diffusion coefficients depend on control, the control set does not need to be convex, the coefficients of jump terms are independent of control as well as the coefficients of mean-field backward stochastic differential equations depend on the joint law of (X(t), Y (t)). Since the coefficients depend on measure, higher mean-field terms could be involved. In order to analyse them, two new adjoint equations are brought in and several new generic estimates of their solutions are investigated. Utilizing these subtle estimates, the second-order expansion of the cost functional, which is the key point to analyse the necessary condition, is obtained, and where after the stochastic maximum principle. An illustrative application to mean-field game is considered.

Controlled Singular Volterra Integral Equations and Pontryagin Maximum Principle

This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for optimal controls. Different from the existing literature, our method enables us to deal with the problem without assuming regularity conditions on the controls, the convexity condition on the control domain, and some additional unnecessary conditions on the nonlinear terms of the integral equation and the cost functional.

The Maximum Principle for Progressive Optimal Stochastic Control Problems with Random Jumps

In this paper, we obtain the maximum principle for optimal controls of stochastic systems with jumps by introducing a new method of variation. The control is allowed to enter both diffusion and jump terms and the control domain need not be convex.

Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator

The field of sub-Riemannian geometry has flourished in the past four decades through the strong interactions between problems arising in applied science (in areas such as robotics) and questions of a pure mathematical character about the nature of space. Methods of control theory, such as controllability properties determined by Lie brackets of vector fields, the Hamilton equations associated to the Maximum Principle of optimal control, Hamilton-Jacobi-Bellman equation etc. have all been found to be basic tools for answering such questions. In this paper, we find a useful role for the vantage point of sub-Riemannian geometry in attacking a problem of interest in non-equilibrium statistical mechanics: how does one create rules for operation of micro- and nano-scale systems (heat engines) that are subject to fluctuations from the surroundings, so as to be able to do useful things such as converting heat into work over a cycle of operation? We exploit geometric optimal control theory to produce such rules selected for maximal efficiency. This is done by working concretely with a model problem, the stochastic oscillator. Essential to our work is a separation of time scales used with great efficacy by physicists and justified in the linear response regime.

Maximum principle for optimal control of McKean‐Vlasov FBSDEs with Lévy process via the differentiability with respect to probability law

SummaryIn this paper, we study stochastic optimal control problem for general McKean‐Vlasov–type forward‐backward differential equations driven by Teugels martingales, associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our main result is based on the differentiability with respect to probability law and a corresponding Itô formula.

On Peng's type maximum principle for optimal control of mean-field stochastic differential equations with jump processes

In this paper, we investigate the Peng's type optimal control problems for stochastic differential equations of mean-field type with jump processes. The coefficients of the system contain not only the state process but also its marginal distribution through their expected values. We assume that the control set is a general open set that is not necessary convex. The control variable is allowed to enter into both diffusion and jump terms. We extend the maximum principle of Buckdahn et al. (2011) to jump case.

The Discrete-Time Geometric Maximum Principle

We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. These results are organized around a new theorem on critical and approximate critical points for discrete-time geometric control systems. We show that this theorem can be used to derive Lie group variational integrators in Hamiltonian form; to establish a maximum principle for control problems in the absence of state constraints; and to provide sufficient conditions for exact penalization techniques in the presence of state or mixed constraints. Exact penalization techniques are used to study sensitivity of the optimal value function to constraint perturbations and to prove necessary conditions for optimality, including in the form of a maximum principle, for discrete-time geometric control problems with state or mixed constraints.

Generalized Maximum Principle in Optimal Control

For an optimal control problem, the concept of a strong local infimum is introduce, for which necessary conditions consisting of some family of principles are formulated. If a function delivers a strong local minimum in this problem (and therefore, a~strong local infimum), then this family contains the classical Pontryagin maximum principle. As a corollary, we derive generalized necessary conditions for a strong local minimum for a problem of the calculus of variations. Examples are given to show that the necessary conditions obtained in the present paper generalize and strengthen classical results.

Generalized Maximum Principle in Optimal Control

The concept of a local infimum for an optimal control problem is introduced, and necessary conditions for it are formulated in the form of a family of “maximum principles.” If the infimum coincides with a strong minimum, then this family contains the classical Pontryagin maximum principle. Examples are given to show that the obtained necessary conditions strengthen and generalize previously known results.