Infinite Dimensional Optimal Control Problem Research Articles

Numerical Methods for Fractional Optimal Control and Estimation

Fractional calculus has become a valuable mathematical tool for modeling various physical phenomena exhibiting anomalous dynamics such as memory and hereditary properties. However, the fractional operators lead to difficulties in analysis, optimization, and estimation that limit the application of fractional models. This paper develops numerical methods to solve fractional optimal control and estimation problems with Caputo derivatives of arbitrary order.
 First, fractional Pontryagin's maximum principle is used to formulate first-order necessary conditions for fractional optimal control problems. A fractional collocation method using polynomial basis functions is then proposed to discretize the resulting boundary value problems. This allows transforming an infinite-dimensional optimal control problem into a finite nonlinear programming problem.
 Second, for fractional estimation, a novel ensemble Kalman filter is proposed based on a Monte Carlo approach to propagate the fractional state dynamics. This provides a recursive fractional state estimator analogous to the classical Kalman filter.
 The capabilities of the proposed collocation and ensemble Kalman filter methods are demonstrated through applications including fractional epidemic control, thermomechanical oscillator control, and state estimation of viscoelastic mechanical systems. The results illustrate improved accuracy over prior discretization schemes along with the ability to handle complex system dynamics.
 This work provides a comprehensive framework for numerical solution of fractional optimal control and estimation problems. The methods enable applying fractional calculus to address challenges in robotics, biomedicine, mechanics, and other fields where systems exhibit non-classical dynamics.

A Necessary Optimality Condition for Optimal Control of Caputo Fractional Evolution Equations

In this paper, we prove the Pontryagin maximum principle, which constitutes the necessary optimality condition, for the infinite-dimensional optimal control problem of X-valued left Caputo fractional evolution equations, where X is a Banach space. An important step in the proof to obtain the desired Hamiltonian maximization condition is to establish new variational and duality analysis. While the former is characterized by a linear X-valued left Caputo fractional evolution equation via spike variation, the latter requires the adjoint equation characterized by a linear X*-valued right Riemann-Liouville (RL) fractional evolution equation, where X* is a dual space of X. We show the variational and duality analysis with the help of the infinite-dimensional fractional version of the technical lemma and the explicit representation of solutions to linear (Caputo and RL) fractional evolution equations using left and right RL state-transition evolution operators.

A Stochastic Model of Economic Growth in Time-Space

We deal with an infinite horizon, infinite dimensional stochastic optimal control problem arising in the study of economic growth in time-space. Such a problem has been the object of various papers in deterministic cases when the possible presence of stochastic disturbances is ignored (see, e.g., [P. Brito, The Dynamics of Growth and Distribution in a Spatially Heterogeneous World, working paper 2004/14, ISEG-Lisbon School of Economics and Management, University of Lisbon, 2004], [R. Boucekkine, C. Camacho, and G. Fabbri, J. Econom. Theory, 148 (2013), pp. 2719--2736], [G. Fabbri, J. Econom. Theory, 162 (2016), pp. 114--136], and [R. Boucekkine, G. Fabbri, S. Federico, and F. Gozzi, J. Econom. Geography, 19 (2019), pp. 1287--1318]). Here we propose and solve a stochastic generalization of such models where the stochastic term, in line with the standard stochastic economic growth models (see, e.g., the books [A. G. Malliaris and W. A. Brock, Stochastic Methods in Economics and Finance, Advanced Textbooks in Economics 17, North Holland, 1982, Chapter 3] and [H. Morimoto, Stochastic Control and Mathematical Modeling: Applications in Economics, Cambridge Books, 2010, Chapter 9]), is a multiplicative one, driven by a cylindrical Wiener process. The problem is studied using the dynamic programming approach. We find an explicit solution of the associated HJB equation, use a verification type result to prove that such a solution is the value function, and find the optimal feedback strategies. Finally, we use this result to study the asymptotic behavior of the optimal trajectories.

Optimal Portfolio Choice With Path-Dependent Labor Income: Finite Retirement Time

This paper extends the project initiated in and studies a lifecycle portfolio choice problem with borrowing constraints and finite retirement time in which an agent receives labor income that adjusts to financial market shocks in a path dependent way. The novelty here, with respect to, is the fact that we have a finite retirement time, which makes the model more realistic, but harder to solve. The presence of both path-dependency, as in, and finite retirement, leads to a two-stage infinite dimensional stochastic optimal control problem, a family of problems which, to our knowledge, has not yet been treated in the literature. We solve the problem completely, and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if state constraints are present. Note that, differently from, here the HJB equation is of parabolic type, hence the work to identify the solutions and optimal feedbacks is more delicate, as it involves, in particular, time-dependent state constraints, which, as far as we know, have not yet been treated in the infinite dimensional literature. The explicit solution allows us to study the properties of optimal strategies and discuss their financial implications.

Trajectory Optimization of Autonomous Agents With Spatio-Temporal Constraints

This article addresses the problem of optimally controlling trajectories of autonomous mobile agents (e.g., robots) so as to jointly minimize travel time and energy consumption in the presence of multiple spatio-temporal constraints on these trajectories. In addition to state and input constraints, we impose spatial equality and temporal inequality constraints viewed as interior-point constraints. We address this problem by first identifying the structure of the optimal agent controllable acceleration profile and showing that it is characterized by several parameters subsequently used for trajectory design optimization. Therefore, the infinite dimensional optimal control problem is transformed into a finite dimensional parametric optimization problem. The proposed algorithm is applied to the eco-driving problem of autonomous vehicles approaching multiple signalized intersections. We include simulation results to show quantitatively the advantages of the proposed solution.

Optimal Portfolio Choice with Path Dependent Labor Income: the Infinite Horizon Case

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agent receives labor income which adjusts to financial market shocks in a path dependent way. This path dependency is the novelty of the model and leads to an infinite dimensional stochastic optimal control problem. We solve the problem completely and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional HJB equation, even if state constraints are present. To the best of our knowledge, this is the first infinite dimensional generalization of Merton's optimal portfolio problem for which explicit solutions can be found. The explicit solution allows us to study the properties of optimal strategies and discuss their financial implications.

Optimal Portfolio Choice with Path Dependent Labor Income: The Infinite Horizon Case

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agent receives labor income which adjusts to financial market shocks in a path dependent way. This path-dependency is the novelty of the model, and leads to an infinite dimensional stochastic optimal control problem. We solve the problem completely, and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if state constraints are present. To the best of our knowledge, this is the first infinite dimensional generalization of Merton's optimal portfolio problem for which explicit solutions can be found. The explicit solution allows us to study the properties of optimal strategies and discuss their financial implications.

Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula

Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper, we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal control problems. The method applies in the case of additively controlled Ornstein–Uhlenbeck processes, when the associated Hamilton–Jacobi–Bellman (HJB) equation admits a mild solution (in the sense of [J. Differential Equations 262 (2017) 3343–3389]). The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature (see, e.g., [Comm. Partial Differential Equations 20 (1995) 775–826]), that the mild solution is a strong solution, that is, a suitable limit of classical solutions of approximating HJB equations. To achieve the goal, we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.

Solving Stochastic LQR Problems by Polynomial Chaos

We consider the infinite dimensional stochastic linear quadratic optimal control problem for the infinite horizon case. We provide a numerical framework for solving this problem using a polynomial chaos expansion approach. By applying the method of chaos expansions to the state equation, we obtain a system of deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of infinite horizon deterministic linear quadratic regulator problems. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to the direct approach. We perform numerical experiments which validate our approach and compare the finite and infinite horizon case.

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

In this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed/discontinuous approximation) measured in suitable norms are derived, showing optimal orders of convergence.

Sensitivity analysis of the value function for infinite dimensional optimal control problems and its relation to Riccati equations

First and second order sensitivities of the value function in optimal control problems constrained by semilinear parabolic equations with respect to the initial condition are considered. Sufficient conditions are given which allow to express these sensitivities in terms of the adjoint equation and a suitably defined Riccati equation.

Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method

In this paper, we introduce a novel pseudospectral method for the numerical solution of optimal control problems governed by a parabolic distributed parameter system. The infinite-dimensional optimal control problem is reduced into a finite-dimensional nonlinear programming problem through shifted Gegenbauer quadratures constructed using a stable barycentric representation of Lagrange interpolating polynomials and explicit barycentric weights for the shifted Gegenbauer-Gauss (SGG) points. A rigorous error analysis of the method is presented, and a numerical test example is given to show the accuracy and efficiency of the proposed pseudospectral method.

Method of evolving junctions: A new approach to optimal control with constraints

We propose a new strategy, called method of evolving junctions (MEJ), to compute the solutions for a class of optimal control problems with constraints on both state and control variables. Our main idea is that by leveraging the geometric structures of the optimal solutions, we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points, called junctions. Then, using a modified gradient flow method, whose dimension can change dynamically, we find local solutions for the optimal control problem. We also employ intermittent diffusion, a global optimization method based on stochastic differential equations, to obtain the global optimal solution. We demonstrate, via a numerical example, that MEJ can effectively solve path planning problems in dynamical environments.

Centralized and optimal motion planning for large-scale AGV systems: A generic approach

A centralized multi-AGV motion planning method is proposed. In contrast to the prevalent planners with decentralized (decoupled) formulations, a centralized planner contains no priority assignment, decoupling, or other specification strategies, thus is free from being case-dependent and deadlock-involved. Although centralized motion planning is computationally expensive, it deserves investigations in schemes that are sensitive to solution quality but insensitive to computation time. Specifically, centralized multi-AGV motion planning is formulated as an optimal control problem in this work, wherein differential algebraic equations are used to describe the AGV dynamics, mechanical restrictions, and exterior constraints. Orthogonal collocation direct transcription method is adopted to discretize the original infinite-dimensional optimal control problem into a large-scale nonlinear programming (NLP) problem, which is solved using interior point method thereafter. Exhaustive simulations are conducted on 10-AGV formation reconfiguration tasks. Simulation results show the validation, unification, and real-time implementation potential of the introduced centralized planner. Particularly, the computation time on a PC reduces to several seconds with near-optimal initial guess in the NLP solving process, making receding horizon replanning possible via this centralized planner.

Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control

A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff, respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is defined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest.

Nonseparation of Sets and Optimality Conditions

Two new concepts of uniform approximating cones are introduced and discussed. The main result is a theorem for nonseparability of two closed sets. As an application of this result, an abstract Lagrange multiplier rule and a necessary optimality condition of Pontryagin maximum principle type for an optimal control problem in infinite-dimensional state space are obtained. These results allow one to treat infinite-dimensional optimal control problems with nonconvex target. The proposed approach reveals the importance of the uniformity of a tangent cone for obtaining necessary optimality conditions in an infinite-dimensional setting.

Finite dimensional approximations for a class of infinite dimensional time optimal control problems

ABSTRACTIn this work, we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge (in appropriate norms) to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.

Safe landing area determination for a Moon lander by reachability analysis

In the last decades developments in space technology paved the way to more challenging missions like asteroid mining, space tourism and human expansion into the Solar System. These missions result in difficult tasks such as guidance schemes for re-entry, landing on celestial bodies and implementation of large angle maneuvers for spacecraft. There is a need for a safety system to increase the robustness and success of these missions. Reachability analysis meets this requirement by obtaining the set of all achievable states for a dynamical system starting from an initial condition with given admissible control inputs of the system.This paper proposes an algorithm for the approximation of nonconvex reachable sets (RS) by using optimal control. Therefore subset of the state space is discretized by equidistant points and for each grid point a distance function is defined. This distance function acts as an objective function for a related optimal control problem (OCP). Each infinite dimensional OCP is transcribed into a finite dimensional Nonlinear Programming Problem (NLP) by using Pseudospectral Methods (PSM). Finally, the NLPs are solved using available tools resulting in approximated reachable sets with information about the states of the dynamical system at these grid points. The algorithm is applied on a generic Moon landing mission. The proposed method computes approximated reachable sets and the attainable safe landing region with information about propellant consumption and time.

Optimal Control for Stochastic Delay Evolution Equations

In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.

Shape deformation analysis from the optimal control viewpoint

A crucial problem in shape deformation analysis is to determine a deformation of a given shape into another one, which is optimal for a certain cost. It has a number of applications in particular in medical imaging.In this article we provide a new general approach to shape deformation analysis, within the framework of optimal control theory, in which a deformation is represented as the flow of diffeomorphisms generated by time-dependent vector fields. Using reproducing kernel Hilbert spaces of vector fields, the general shape deformation analysis problem is specified as an infinite-dimensional optimal control problem with state and control constraints. In this problem, the states are diffeomorphisms and the controls are vector fields, both of them being subject to some constraints. The functional to be minimized is the sum of a first term defined as geometric norm of the control (kinetic energy of the deformation) and of a data-attachment term providing a geometric distance to the target shape.This point of view has several advantages. First, it allows one to model general constrained shape analysis problems, which opens new issues in this field. Second, using an extension of the Pontryagin maximum principle, one can characterize the optimal solutions of the shape deformation problem in a very general way as the solutions of constrained geodesic equations. Finally, recasting general algorithms of optimal control into shape analysis yields new efficient numerical methods in shape deformation analysis. Overall, the optimal control point of view unifies and generalizes different theoretical and numerical approaches to shape deformation problems, and also allows us to design new approaches.The optimal control problems that result from this construction are infinite dimensional and involve some constraints, and thus are nonstandard. In this article we also provide a rigorous and complete analysis of the infinite-dimensional shape space problem with constraints and of its finite-dimensional approximations.