Class Of Optimal Control Problems Research Articles

The Time-Scaling Transformation Technique for Optimal Control Problems with Time-Varying Time-Delay Switched Systems

In this paper, we consider a class of optimal control problems where the dynamical systems are time-delay switched systems with the delay being a function of time. By applying the control parameterization method, the control heights and switching times become decision variables that need to be optimized. It is well-known that, for this type problem, the variable switching times cannot be optimized directly. To work around this problem, we introduce a time-scaling transformation technique so that the original system is transformed an equivalent system, which is defined on a new time horizon with fixed switching times. Based on the relationship between the original time scale and the new time scale, we derive the gradients of the objective and constraint functions with respect to the control heights and durations. Then, the new problem can be solved by gradient-based optimization approach. To demonstrate the effectiveness of the time-scaling transformation technique, two example problems are solved.

An Optimal Control Problem Governed by Nonlinear First Order Dynamic Equation on Time Scales

In this paper, we are concerned with a class of optimal control problem governed by nonlinear first order dynamic equation on time scales. By imposing some suitable conditions on the related functions, for any given control policy, we first obtain the existence of a unique solution for the nonlinear controlled system. Then, we study the existence of an optimal solution for the optimal control problem.

Numerical Solutions for the Optimal Control Governing by Variable Coefficients Nonlinear Hyperbolic Boundary Value Problem Using the Gradient Projection, Gradient and Frank Wolfe Methods

This paper is concerned with studying the numerical solution for the discrete classical optimal control problem (NSDCOCP) governed by a variable coefficients nonlinear hyperbolic boundary value problem (VCNLHBVP). The DSCOCP is solved by using the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the NS for the discrete weak form (DWF) and for the discrete adjoint weak form (DSAWF) While, the gradient projection method (GRPM), also called the gradient method (GRM), or the Frank Wolfe method (FRM) are used to minimize the discrete cost function (DCF) to find the DSCOC. Within these three methods, the Armijo step option (ARMSO) or the optimal step option (OPSO) are used to improve the discrete classical control (DSCC). Finally, some illustrative examples for the problem are given to show the accuracy and efficiency of the methods.

Optimal control of an eddy current problem with a dipole source

This paper is concerned with the analysis of a class of optimal control problems governed by a time-harmonic eddy current system with a dipole source, which is taken as the control variable. A mathematical model is set up for the state equation where the dipole source takes the form of a Dirac mass located in the interior of the conducting domain. A non-standard approach featuring the fundamental solution of a curlcurl−Id operator is proposed to address the well-posedness of the state problem, leading to a split structure of the state field as the sum of a singular part and a regular part. The aim of the control is the best approximation of desired electric and magnetic fields via a suitable L2-quadratic tracking cost functional. Here, special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived.

Study of the Dynamic Characteristics of Some Circuit Optimization Strategies

The work explores the possibility of a significant reduction in processor time required to optimize analog circuits. For this, we can reformulate and generalize the problem of chain optimization based on the approaches of optimal control theory. The analog circuit design process is formulated as a dynamic controlled system. A special control vector has been defined to redistribute the computational cost between circuit analysis and parametric optimization. This redistribution significantly reduces CPU time. The task of designing a circuit for the minimum processor time can be formulated as a classical optimal control problem while minimizing some functional. The introduction of the concept of the Lyapunov function of a controlled dynamic system made it possible to use it to analyse the main characteristics of the design process. An analysis of the behaviour of a special function, which is a combination of the Lyapunov function and its time derivative, made it possible to compare various design strategies and select the best ones that are executed in minimal CPU time.

Selection of Control Vector Switching Points for Circuit Optimization

The process of optimizing the circuit is formulated as a controlled dynamic system. A special control vector has been defined to redistribute the computational cost between circuit analysis and parametric optimization. This redistribution will help solve the task of optimizing the circuit for minimum CPU time. In this case, the task of optimizing the circuit with minimal CPU time can be formulated as the classical optimal control problem for minimizing some functional. The concept of the Lyapunov function of a controlled dynamic system is used to analyze the main characteristics of the design process. An analysis of the Lyapunov function and its time derivative makes it possible to predict the optimal structure of the control vector for constructing an optimal or quasi-optimal circuit design algorithm. The results are based on the previously discovered effect of accelerating the design process. In this case, the optimal structure of the control vector is determined, which minimizes processor time.

An efficient algorithm for solving the system optimisation problem in transportation

ABSTRACTIn this paper, the classical optimal control problem in air transportation system is studied. The optimisation model is based on the principle of ‘first come first serve’ and the arrival traffic flow only occurs in the first queue which is assumed to be infinite. In addition, it is assumed that all processing times and arrival intervals are predefined. This paper aims to propose a new optimal control problem of series air traffic system and an effective solution. The exact algorithm of each sub-segment is given, and the required results are obtained. Numerical experiments show the effectiveness of the algorithm.

Optimal control of multiagent systems in the Wasserstein space

This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in $${\mathbb {R}}^d$$ in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing his/her underlying microscopic dynamics, depends both on his/her position, and on the configuration of all the other agents at the same time. So the problem is naturally stated in the space of probability measures on $${\mathbb {R}}^d$$ equipped with the Wasserstein distance. The main result of the paper gives a new characterization of the value function as the unique viscosity solution of a first order partial differential equation. We introduce and discuss several equivalent formulations of the concept of viscosity solutions in the Wasserstein spaces suitable for obtaining a comparison principle of the Hamilton Jacobi Bellman equation associated with the above control problem.

The Continuous Classical Optimal Control governing by Triple Linear Parabolic Boundary Value Problem

This paper deals with the continuous classical optimal control problem for triple partial differential equations of parabolic type with initial and boundary conditions; the Galerkin method is used to prove the existence and uniqueness theorem of the state vector solution for given continuous classical control vector. The proof of the existence theorem of a continuous classical optimal control vector associated with the triple linear partial differential equations of parabolic type is given. The derivation of the Fréchet derivative for the cost function is obtained. At the end, the theorem of the necessary conditions for optimality of this problem is stated and is proved.

The Continuous Classical Optimal Control Problems for Triple Elliptic Partial Differential Equations

In this paper the Galerkin method is used to prove the existence and uniqueness theorem for the solution of the state vector of the triple linear elliptic partial differential equations for fixed continuous classical optimal control vector. Also, the existence theorem of a continuous classical optimal control vector related with the triple linear equations of elliptic types is proved. The existence of a unique solution for the triple adjoint equations related with the considered triple of the state equations is studied. The Fréchet derivative of the cost function is derived. Finally the theorem of necessary conditions for optimality of the considered problem is proved.

Chebyshev cardinal functions for a new class of nonlinear optimal control problems with dynamical systems of weakly singular variable-order fractional integral equations

The main objectives of this study are to introduce a new class of optimal control problems governed by a dynamical system of weakly singular variable-order fractional integral equations and to establish a computational method by utilizing the Chebyshev cardinal functions for their numerical solutions. In this way, a new operational matrix of variable-order fractional integration is generated for the Chebyshev cardinal functions. In the established method, first the control and state variables are approximated by the introduced basis functions. Then, the interpolation property of these basis functions together with their mentioned operational matrix is applied to derive an algebraic equation instead of the objective function and an algebraic system of equations instead of the dynamical system. Eventually, the constrained extrema technique is applied by adjoining the constraints generated from the dynamical system to the objective function using a set of Lagrange multipliers. The accuracy of the established approach is examined through several test problems. The obtained results confirm the high accuracy of the presented method.

Lossless Convexification of Optimal Control Problems with Semi-continuous Inputs

This paper presents a novel one-shot convex optimization method for finding globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the input norms to be either zero or lower-and upper-bounded. The non-convex problem is relaxed to a convex one whose optimal solution is proved to be optimal almost everywhere for the original problem, a procedure known as lossless convexification. The solution relies on second-order cone programming and demonstrates that a meaningful class of optimal control problems with binary variables can be solved reliably and in polynomial time. A rocket landing example with a coupled thrust-gimbal constraint corroborates the effectiveness of the approach.

Bài toán điều khiển phân bố và điều khiển biên cho phương trình đạo hàm riêng elliptic nửa tuyến tính

In this paper we investigate existence of solutions, optimality conditions, and solution stability for a class of optimal control problems governed by semi linear elliptic partial differential equations. In the class of optimal control problems, distributed controls and boundary controls will be considered, they may appear nonlinearly in the state equation. This class of control problems is more general and complicated, investigation of them is interesting and meaningful.

Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data

We consider a class of optimal control problems in which the cost to minimize comprises both a final cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the final cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involves the concepts of lower Dini derivative and the proximal normal vectors; if the final cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.

On Optimal Control of Discounted Cost Infnite-Horizon Markov Decision Processes Under Local State Information Structures

This paper investigates a class of optimal control problems associated with Markov processes with local state information. The decision-maker has only a local access to a subset of a state vector information as often encountered in decentralized control problems in multi-agent systems. Under this information structure, part of the state vector cannot be observed. We leverage ab initio principles and find a new form of Bellman equations to characterize the optimal policies of the control problem under local information structures. The dynamic programming solutions feature a mixture of dynamics associated unobservable state components and the local state-feedback policy based on the observable local information. We further characterize the optimal local-state feedback policy using linear programming methods. To reduce the computational complexity of the optimal policy, we propose an approximate algorithm based on virtual beliefs to find a sub-optimal policy. We show the performance bounds on the sub-optimal solution and corroborate the results with numerical case studies.

Design of LQR Tracking Controller Combined With Orthogonal Collocation State Planning for Process Optimal Control

As one of the most important optimization methods for process optimal control, orthogonal collocation method has been widely used. However, this kind of optimization method is generally an open-loop optimization frame, which makes the model disturbances greatly affect the control performance and quality. In order to improve the optimal control performance of dynamic systems with disturbances, this work proposed an orthogonal collocation optimization-based linear quadratic regulator (OC-LQR) control method for process optimal control problems. Firstly, the orthogonal collocation method is derived in detail to transform the optimal control problem so that optimal state curves can be accordingly calculated. Then, an improved LQR controller design is proposed by using the obtained optimal state curves so as to construct the feedback control frame. On this basis, a state planning-based LQR tracking control is established with closed-loop control characteristics to tackle model disturbance problem. Meanwhile, the detailed Simulink model of the proposed control method is constructed for simulation execution. Finally, the proposed control structure is tested on a classical process optimal control problem with model disturbance (white noise) test and Gaussian mixture noise test to verify the performance of the proposed method. Simulation studies show that the control performance of the proposed method is excellent, where the mean absolute error of state curve tracking averagely reduces by 34.73% and the minimal performance index improves by 64.44% when compared with the classical orthogonal optimal control method. Simulation results demonstrate the effectiveness of the proposed planning-based tracking control framework.

Optimal control of a black-box system based on surrogate models by spatial adaptive partitioning method

When the dynamic model of a classical optimal control problem is explicit, we can transform this problem into a nonlinear programming problem and solve it by employing a traditional method. However, in some cases, no mathematical model of state equations is provided explicitly except for input–output data obtained from a simulation model. The hybrid model composed of functional mockup unit blocks generated in multiple platforms is a typical example. In this work, we regard these blocks as black-box models and use hierarchical neural network model to surrogate right-hand-side derivative functions of state equations. Specifically, to obtain highly accurate hierarchical neural network model, we explore a spatial adaptive partitioning criterion combining global sensitivity indices and interval length of local spaces based on the input–output data. Compared with models trained by several other partition criteria, numerical results verify that surrogate models obtained by the spatial adaptive partitioning method have higher accuracy. A mathematical example and a trajectory optimization problem of the black-box industrial robot Manutec r3 indicate the effectiveness of our proposed strategy.

Necessary Optimality Conditions in Isoperimetric Constrained Optimal Control Problems

In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, necessary optimality conditions are established for a feasible solution in the considered optimization problem. More precisely, under simplified hypotheses and using a modified Legendrian duality, we establish a maximum principle for the considered optimization problem.

Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

AbstractIn this paper, the optimal control problems (OCPs) are converted into a constrained optimization problem based on state parameterization via the B‐spline functions (BSFs). In fact, the state variable can be considered as a series of the BSFs with unknown coefficients, and the OCPs are transformed into a constrained optimization problem. With the proposed method, the control and state variables also the performance index can be obtained approximately. Also, the convergence theorem of the presented approach is proved in details, some illustrative examples are reported. Also, an example, which has analytic noncontinuous state and control variable, is presented to show the efficiency and reliability of the purposed method, compared with other existing methods.

A time-scaling technique for time-delay switched systems

In this paper, we consider a class of optimal control problems governed by nonlinear time-delay switched systems, in which the system parameters and switching times between different subsystems are decision variables to be optimized. We propose a new computational approach to deal with the computational difficulties caused by variable switching times. The original time-delay switched system is firstly transformed into an equivalent switched system defined on a new time horizon where the switching times are fixed, but each of the subsystems contain a variable time-delay that depends on the durations of each sub-system in the original system. By deriving the analytical form for the variable time-delay in the new time horizon, we can solve the new time-delay switched system. Then, gradient-based optimization algorithm can be applied to solve the equivalent problem efficiently. Numerical results show that this new approach is effective.