Class Of Optimal Control Problems Research Articles

On the regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems

A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the so-called generalized separation condition is satisfied, then both optimality conditions of KKT-type and regularity of multipliers are fulfilled. Moreover, we show that if the initial value is good enough and boundary ∂Ω has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.

Quaternary Continuous Classical Boundary Optimal Control Problem Dominating by Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem dominated by a quaternary linear parabolic boundary value problem. Under suitable assumptions and with a given quaternary continuous classical boundary control vector, the existence theorem for a unique quaternary state vector solution to the weak form is stated and demonstrated via the method of Galerkin. Furthermore, the continuity of the Lipschitz operator between the state vector solution to the weak form for the dominating equations and the corresponding are proved. The existence of a quaternary continuous classical boundary optimal control vector is stated and demonstrated under suitable Assumptions. The mathematical formulation for the quaternary adjoint boundary value problem associated with each considered boundary value problem is obtained and the Frѐchet derivative for the objective function is derived. Finally, the necessary conditions for the optimality theorem of the problem are stated and demonstrated.

Bellman Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

This work introduces Bellman Neural Networks (BeNNs) and employs them to learn the optimal control actions for the class of optimal control problems (OCPs) with integral quadratic cost. BeNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed and trained to tackle OCPs via applying the Bellman Principle of Optimality (BPO). The BPO provides necessary and sufficient optimality conditions, which result in a nonlinear partial differential equation known as the Hamilton-Jacobi-Bellman (HJB) equation. BeNNs learn the optimal control actions from the unknown solution of the arising HJB equation (i.e., the value function), where the unknown solution is modeled using a Neural Network. Additionally, the paper shows how to estimate the upper bounds on the generalization error of BeNNs while learning the solutions for the OCP class under consideration. The generalization error estimate is provided in terms of the choice and number of the training points as well as the training error. Numerical studies show that BeNNs can be successfully applied to learn the feedback control actions for the class of optimal control problems considered and, after the training is completed, deployed to control the system in a closed-loop fashion. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Impact Statement</i> —The proposed research improves our understanding of how to solve optimal control problems with closed-loop solutions and has potentially a countless number of applications in several different areas. The study is at the intersection between optimal control theory and artificial intelligence connected with mathematical tools for functional interpolation. This advances the ability to implement a higher level of autonomy in decision-making for practical applications with a beneficial impact on our society.

Cooperative Eco-Driving Control of Connected Multi-Vehicles With Spatio-Temporal Constraints

In this article, we propose a novel time-energy optimal control approach with applications in cooperative eco- driving of connected and automated vehicles (CAVs) in urban traffic networks. Safely approaching and departing signalized intersections requires the satisfaction of both spatial equality constraints determined by intersection locations and temporal inequality constraints in compliance with the green light phases. To generate time- and energy-optimal trajectories, the optimal crossing times at intersections are firstly treated as characteristic time constraints, which makes the problem tractable. Then the di- rect multiple shooting method and time transformation technique are applied to find a numerical solution. The contribution of this article is twofold. The first one is the development of a novel time- and energy-optimal control approach that ensures a trade- off between minimizing energy and time for a general class of optimal control problems with multiple characteristic times. The second contribution is the application of the proposed method to the challenging problem of multi-CAVs’ cooperative eco-driving control, in which multiple vehicles must simultaneously minimize travel time and energy consumption in the presence of spatio- temporal constraints. Simulation analysis over real-world urban route scenarios shows that the proposed eco-driving control strategy can save up to 8.2% of energy or reduce up to 6.7% of travel time compared to a baseline method. Furthermore, hardware-in-the-loop (HiL) experimental results indicate that the proposed strategy can be implemented in real-time.

An improved slope-based adaptive control vector parameterization method for dynamic programming

Control vector parameterization method is the most commonly used numerical computation method in solving dynamic programming problems. However, to balance the expected trajectory and the computational cost, this method faces difficulties in dividing the optimal discretization time grid. In this paper, we propose an effective control approach for non-uniform adaptive grid division. By analyzing the slope change trend of the control parameter, time nodes are adaptively refined by merging time grids with gentle slopes to remove unnecessary time nodes and adding time nodes to time grids with steep slopes to improve function approximation accuracy. Eventually, we obtain an adaptive time grid division method under which the trajectory of discretization control parameter is more approximate to the optimal control trajectory. Under the condition of intuitive slope information, the proposed method can achieve better performance with fewer optimization parameters and shorter computation time. Finally, the proposed method is applied to solve a classic optimal control problem and the obtained results are compared with the traditional control vector parameterization method. By comparison through the example, our proposed method can overcome the contradiction between approximation accuracy and computational cost in control vector parameterization method and improve the optimization efficiency.

Relaxation approach for learning neural network regularizers for a class of identification problems

The present paper deals with the data-driven design of regularizers in the form of artificial neural networks, for solving certain inverse problems formulated as optimal control problems. These regularizers aim at improving accuracy, wellposedness or compensating uncertainties for a given class of optimal control problems (inner-problems). Parameterized as neural networks, their weights are chosen in order to reduce a misfit between data and observations of the state solution of the inner- optimal control problems. Learning these weights constitutes the outer-problem. Based on necessary first-order optimality conditions for the inner-problems, a relaxation approach is proposed in order to implement efficient solving of these inner-problems, namely the forward operator of the outer-problem. Optimality conditions are derived for the latter, and are implemented in numerical illustrations dealing with the inverse conductivity problem. The numerical tests show the feasibility of the relaxation approach, first for rediscovering standard L 2-regularizers, and next for designing regularizers that compensate unknown noise on the observed state of the inner-problem.

A Class of Optimal Control Problems of Forward–Backward Systems with Input Constraint

A Class of Optimal Control Problems of Forward–Backward Systems with Input Constraint

A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems

AbstractIn this work, we introduce a Coupled Radial Basis Functions collocation (CRBFs‐Coll) approach for solving optimal control problems. CRBFs are real‐valued Radial Basis Functions (RBFs) augmented with a conical spline. They show insensitivity to the shape parameter resulting in robust behavior in function approximation. The method is applied to classical nonlinear optimal control problems: Zermelo's problem, a Duffing oscillator with various boundary conditions, and a nonlinear pendulum on a cart problem. The nonlinear optimal control problem is solved by deriving the necessary conditions for optimality followed by collocation using CRBFs as basis functions for approximating the two‐point boundary value problem (TPBVP). The resulting system of nonlinear algebraic equations (NAEs) is then solved using a standard solver. Unlike existing methods that rely on nodal distributions that are denser at the boundaries to obtain a solution, the present CRBFs‐Coll approach is capable of solving the optimal control problem on uniform nodes without the need for interpolation. The present CRBFs‐Coll approach is shown to be simple to implement and provides accurate results over an equidistant nodal distribution while maintaining a continuous representation of the control and states.

Stochastic maximum principle for recursive optimal control problems with varying terminal time

This paper introduces a new recursive stochastic optimal control problem driven by a forward-backward stochastic differential equations (FBSDEs), where the terminal time varies according to the constraints of the state of the forward equation. We can use this optimal control problem to describe the investment portfolio problems with the varying investment period. Based on novel ρ-moving variational and adjoint equations, we establish the stochastic maximum principle for this optimal control problem including the classical recursive optimal control problem as a particular case. Furthermore, we propose an example to verify the main results of this study.

Optimal control in linear-quadratic stochastic advertising models with memory

AbstractThis paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating an optimization problem equivalent to the original one in a Hilbert space. Such technique, however, requires the Volterra kernel from the forward equation to have a representation of a particular form that may be challenging to obtain in practice. We overcome this issue for Hölder continuous kernels by approximating them with Bernstein polynomials, which turn out to enjoy a simple representation of the required type. Then we solve the optimal control problem for the forward process with approximated kernel instead of the original one and study convergence. The approach is illustrated with simulations.

Optimal control problems in transport dynamics with additive noise

Motivated by the applications in leader-follower multi-agent dynamics, a class of optimal control problems is investigated, where the goal is to influence the behavior of a given population through another controlled one interacting with the first. The evolution of the systems obeys a non-linear Fokker–Planck-type equation as diffusive terms accounting for randomness in the evolution are taken into account. A well-posedness theory under very low regularity of the control vector fields is developed, as well as a rigorous mean-field limit of a stochastic particle approximation.

A collocation method using generalized Laguerre polynomials for solving nonlinear optimal control problems governed by integro-differential equations

The Laguerre polynomials represent a new set of basic functions adopted to solve a class of optimal control problems (OCP) specified by integro-differential equation namely IOCP. The corresponding operational matrices of derivatives are computed to extend the solution of the problem in terms of Laguerre polynomials. Using the results of the considered basis functions with a joint application of collocation method, the IOCP is reduced to solve a system of nonlinear algebraic system. Come acrossing an error bound and convergence analysis for the approximate optimal value of the performance index have been proved by the proposed method. Finally, some examples are given to bring about the validity and applicability of this technique.

A note on the minimum‐norm in dual space approach to some classical linear optimal control problems

AbstractSome classical optimal control problems of linear systems can be characterized as finding minimum‐norm vectors from within linear varieties in appropriate dual spaces. Then, the solution to such an optimal control problem can be derived from the alignment between the optimal vector in the dual space and the optimal vector in the primal space, and the dual maximization problem of the minimum‐norm problem. This note presents a detailed introduction to this minimum‐norm in dual space approach by examples of minimum‐supremum‐norm, minimum‐energy, and minimum‐time optimal control problems of linear systems. Connections and differences between these problems in light of the introduced approach are discussed.

The cost differential for an optimal train journey on level track

The classic optimal train control problem is to drive a train on a track with known gradient over a fixed distance and within a specified time in such a way as to minimize tractive energy consumption. On level track the optimal strategies take two basic forms—a truncated strategy of optimal type with phases of maximum acceleration, coast and maximum brake which is typical of shorter metropolitan journeys, and an extended strategy of optimal type with phases of maximum acceleration, speedhold at the optimal driving speed, coast to the optimal braking speed, and maximum brake which is typical of longer journeys by freight trains and intercity passenger trains. The cost of these optimal strategies is uniquely determined by the journey distance and journey time. In this paper we extend a previously known formula for the partial rate of change of cost with respect to journey time to a formula for the full rate of change of cost that also incorporates the partial rate of change of cost with respect to journey distance.

Relationship Between the Maximum Principle and Dynamic Programming for Minimax Problems

This paper is concerned with the relationship between the maximum principle and dynamic programming for a large class of optimal control problems with maximum running cost. Inspired by a technique introduced by Vinter in the 1980s, we are able to obtain jointly a global and a partial sensitivity relation that link the coextremal with the value function of the problem at hand. One of the main contributions of this work is that these relations are derived by using a single perturbed problem, and therefore, both sensitivity relations hold, at the same time, for the same coextremal. As a by-product, and thanks to the level-set approach, we obtain a new set of sensitivity relations for Mayer problems with state constraints. One important feature of this last result is that it holds under mild assumptions, without the need of imposing strong compatibility assumptions between the dynamics and the state constraints set.

REFINEMENT OF OPTIMAL CONTROL PROBLEM FOR PRACTICAL IMPLEMENTATION OF ITS SOLUTION

The solution of the optimal control problem in the classical formulation is control in the form of a function of time. The implementation of such a solution leads to an open control system and therefore cannot be applied directly in practice. It is believed that solving the classical optimal control problem leads to an optimal control program and program trajectory in state space. To implement the movement of the control object along the program trajectory, it is necessary to build an additional movement stabilization system. The problem of synthesizing a system for stabilizing movement along a program trajectory and the requirements that this system should meet do not arise from the classical setting of the optimal control problem. An updated statement of the optimal control problem is given, which includes an additional requirement for an optimal trajectory, and the solution of which can be directly applied in practice in a real control object.

Multi-Aircraft Flight Conflict Resolution and Trajectory Recovery Scheme Based on Mixed Integer Linear Programming and Geometric Rules

A conflict resolution and trajectory recovery scheme based on heading angle change (HAC) is proposed to solve mid-term conflict between multi-aircraft cruising in a shared airspace. Flight conflict resolution has always been regarded as one of the most important responsibilities of air traffic controllers (ATCOs). The proposed conflict resolution scheme can be applied to decision support tools to assist ATCOs in making decisions. The goals of reducing the possibility of human error and increasing the efficiency of airspace operations can be achieved. Firstly, the flight conflict problem is modeled as a class of optimal control problem. Secondly, the optimal solution can be obtained by mixed integer linear programming (MILP) method based on HAC strategy. Thirdly, a trajectory recovery method based on geometric rules is proposed. It can be found through simulated experiment that the extra flight distance and delay adopting the designed scheme is smaller than the compared method.

Turnpike phenomenon for a class of optimal control problems with a Lyapunov function

ABSTRACT In this paper we prove several turnpike results for trajectories of discrete disperse dynamical systems introduced in 1980 by A. M. Rubinov, which have a prototype in mathematical economics.

An Algorithm to Construct Subsolutions of Convex Optimal Control Problems

We propose an algorithm that produces a nondecreasing sequence of subsolutions for a class of optimal control problems distinguished by the property that the associated Bellman operators preserve convexity. In addition to a theoretical discussion and proofs of convergence, numerical experiments are presented to illustrate the feasibility of the method.

A general optimization framework for dynamic time warping

The goal of dynamic time warping is to transform or warp time in order to approximately align two signals. We pose the choice of warping function as an optimization problem with several terms in the objective. The first term measures the misalignment of the time-warped signals. Two additional regularization terms penalize the cumulative warping and the instantaneous rate of time warping; constraints on the warping can be imposed by assigning the value $$+\infty$$ to the regularization terms. Different choices of the three objective terms yield different time warping functions that trade off signal fit or alignment and properties of the warping function. The optimization problem we formulate is a classical optimal control problem, with initial and terminal constraints, and a state dimension of one. We describe an effective general method that minimizes the objective by discretizing the values of the original and warped time, and using standard dynamic programming to compute the (globally) optimal warping function with the discretized values. Iterated refinement of this scheme yields a high accuracy warping function in just a few iterations. Our method is implemented as an open source Python package GDTW.