Optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element analysis

This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find conditions which guarantee that the considered optimal control problem is well posed. Then, when the optimal control problem is well posed, it is natural to look for conditions which guarantee the attainability of the optimal control problem that is being evaluated. The main tool involved in the solution of the problems stated before is a backward jump matrix linear differential equation (BJMLDE) with a Riccati-type jumping operator. This is formulated using the matrix coefficients of the controlled system and the weight matrices of the performance criterion. We show that the well posedness of the optimal control problem under investigation is guaranteed by the existence of the maximal and bounded solution of the associated BJMLDE with a Riccati-type jumping operator. Further, we show that when the associated BJMLDE with a Riccati-type jumping operator has a maximal solution which satisfies a suitable sign condition, then the optimal control problem is attainable if and only if it has an optimal control in a state feedback form, or if and only if the maximal solution of the BJMLDE with a Riccati-type jumping operator is a stabilizing solution. In order to make the paper more self-contained, we present a set of conditions that correspond to the existence of the maximal solution of the BJMLDE satisfying the desired sign condition.

On the stochastic linear quadratic control problem with piecewise constant admissible controls

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

This article introduces the hybridizable discontinuous Galerkin (HDG) approach to numerically approximate the solution of a linear hyperbolic integro-differential equation. A priori error estimates for semi-discrete and fully discrete schemes are developed. It is shown that the optimal order of convergence is achieved for both scalar and flux variables. To achieve that, an intermediate projection is introduced for the semi-discrete error analysis, and it also shows that this projection achieves convergence of order $h^{k+3/2}$ for $k ≥ 1.$ Next, superconvergence is achieved for the scalar variable using element-by-element post-processing. For the fully discrete error analysis, the central difference scheme and the mid-point rule approximate the derivative and the integral term, respectively. Hence, the second order of convergence is achieved in the temporal direction. Finally, numerical experiments have been performed to validate the theory developed in this article.

A sequence of optimal control problems for systems governed by linear hyperbolic equations with the nonhomogeneous Neumann boundary conditions is considered. The integral cost functionals and the differential operators in the equations depend on the param

In this paper, an iterative computational method is proposed to handle constrained nonlinear quadratic optimal control problems. with terminal state constraints, saturation state and saturation control variables constraints. The method is based on replacing the original constrained nonlinear optimal control problem by a sequence of constrained linear time-varying quadratic optimal control problems. To this end, an iterative technique is used to replace the original constrained nonlinear dynamic system by a sequence of constrained linear time-varying systems. Then each of constrained linear time-varying quadratic optimal control problems is approximated by a quadratic programming problem by parameterizing each of the state variable by a finite length Legendre polynomials with unknown parameters. To show the effectiveness of the proposed method, simulation results of a constrained nonlinear optimal control problem are presented.

An integral state-constrained optimal control problem governed by an elliptic partial differential equation and its finite element approximation are considered. The finite element approximation is constructed on multimeshes. An $L^2$-norm a priori error estimate of the finite element approximation is obtained. Further, some superconvergence results are proved. Based on these superconvergence results, almost optimal $L^\infty$-norm error estimates are derived. Some recovery algorithms are then proposed to produce a posteriori error estimators of gradient type. To solve the finite element system, a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence rate is proved. Some numerical examples are performed to confirm theoretical analysis.

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

This paper presents a survey of the most recent literature developments in the field of computational guidance and control for space applications. In recent years, the space sector has changed significantly, shifting towards solutions requiring increased performance and autonomy. This change has also been accelerated by the technical and economic needs of the new space approach. However, traditional guidance and control techniques cannot always meet the increasing requirements of autonomy and performance. For this reason, computational guidance and control has become of increasing interest to the aerospace sector. In contrast to traditional guidance and control, computational guidance and control generates commands in real time by using on-board numerical computation. It represents, therefore, a fundamental step on the way to system autonomy and autonomous operations, with the potential to increase system capabilities and reduce operational costs. On-board computation introduces, however, a series of challenges, including the need to develop algorithms which are computationally efficient, reliable and robust. The literature has classified computational guidance and control techniques into two major categories, namely model-based and data-based. The model-based category encompasses all techniques in which the guidance and control laws are explicitly related to the system dynamic and kinematic models. On the contrary, data-based techniques infer guidance and control laws from a set of data, derived from the models, but do not explicitly use such models in the algorithm itself. Model-based guidance and control includes optimal control, model predictive control and convex optimisation. For the specific case of space attitude re-orientation manoeuvres, additional techniques are path planning and artificial potential functions. The general concept behind the Model Predictive Control (MPC) framework is to solve an optimal control problem repeatedly, for example whenever new measurements are available. The MPC implements only the first element of the returned solution, and the process is then repeated. By doing so, the system is better able to cope autonomously with model uncertainties, failures, or errors. For spacecraft with limited on-board capability, having to repeatedly solve an optimisation problem on-board could represent a challenge. In this case, explicit implementations of the MPC have been proposed, in which the guidance and control laws are precomputed offline and implemented on-board using look-up tables or simple functions. If the optimal control problem of the MPC is convex, or if it can be convexified, convex optimisation can be used directly. If the problem cannot be convexified, the solution can be found by sequentially solving a convex approximation of the original non-convex problem. Convex optimisation can be used either repeatedly in the context of an MPC framework, or as a single function call to generate guidance and control laws, e.g. at the beginning of an attitude re-orientation manoeuvre. Convex optimisation has received increased attention in recent years, also due to its suitability to on-board implementation. In fact, if the optimal control problem is convex, it can be solved reliably and efficiently on-board. Convex optimisation has been successfully applied to the space sector with the notable examples of the SpaceX rocket booster landing and of the Mars pinpoint landing demonstration aboard the Masten Xombie sounding rocket. Model-based guidance techniques, not based on optimisation, have been increasingly explored in recent years, as guidance strategies employed in other engineering fields, such as robotics, are being applied to spacecraft. Among these strategies, of particular importance are path planning and artificial potential field methods, due to their applicability to spacecraft autonomous attitude manoeuvres. Path planning algorithms have been applied for decades to ground vehicles and robotic manipulators. These algorithms consist of techniques to find the best route between two points of a graph representing the search space of a problem. For spacecraft re-orientation applications, the literature has proposed different attitude representation conventions and various cost functions associated to the graph search algorithm. Each approach comes with advantages and disadvantages, which will be analysed in the paper. Along with path planning, Artificial Potential Function (APF) techniques have been developed in the field of robotics, and are now also considered for space applications, to track or maintain a specific reference under a set of constraints. The main idea behind these methods is to synthetise the controller using a potential function. This function is specifically built to grant asymptotic stability to the system, according to Lyapunov’s second stability theorem. This formulation grants flexibility to accommodate system constraints, in a computationally light-weight and efficient manner. Data-based guidance and control laws are mainly based on Artificial Intelligence (AI) technologies, and generate guidance commands after training an algorithm with a large set of data. The interest of the space sector towards AI has accelerated in the last years due to the greater maturity of the technology. Some of the most powerful examples lie in the area of machine learning, like reinforcement learning, supervised learning and deep learning. These algorithms are used to extract patterns from large amounts of data derived from complex nonlinear dynamical and kinematical models. Examples of such data can be numerical solutions of an optimal control problem. The patterns extracted from the data allow the system to identify, in a short computational time, the critical parameters needed to relate the state and control variables. Creating a sufficiently large dataset for training the algorithm requires a considerable computational effort, though, and for certain methods optimality cannot be guaranteed due to the lack of theoretical support derived from the optimality conditions. Advancements made in the last years span several research directions and open a wide variety of opportunities and challenges, which need to be examined to guide further exploration. In this paper, both model-based and data-based techniques and applications are presented. The considered techniques include model predictive control, convex optimisation and machine learning. In addition, for the implementation of autonomous attitude manoeuvres, path planning methods and artificial potential function methods are analysed. These techniques have been applied, in the literature, to a range of different applications. The applications discussed in this paper include, in addition to attitude manoeuvres, powered descent landing guidance, low-thrust orbital transfers, proximity operations and station keeping.

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

We look at L ∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control problem is obtained by adjusting of the temperature on the boundary. We analyze finite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.

In this paper, we study an a priori error analysis for the quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. A priori error estimates for the mixed finite element approximation of nonlinear optimal control problems are obtained. Some numerical examples are presented to confirm our theoretical results.

Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems