Article
Mathematics, Applied
Eduardo Casas, Karl Kunisch, Mariano Mateos
Summary: This paper studies the numerical approximation of an optimal control problem with a constraint on the spatial L-1 norm of the control. The controls are discretized using piecewise constant and continuous piecewise linear functions. The results show that under finite element approximations, the sparsity properties of the continuous solutions can be preserved using piecewise constant approximations of the control, but suitable numerical integration is needed to maintain the sparsity pattern when using spatially continuous piecewise linear approximations. Error estimates and numerical examples are also provided.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Chunjuan Hou, Zuliang Lu, Xuejiao Chen, Fei Huang
Summary: This paper investigates variational discretization for the optimal control problem with nonlinear parabolic equations and control constraints, achieving improved error estimates compared to standard finite element methods with backward Euler. Additionally, the study presents a posteriori error estimates of residual type.
Article
Mathematics, Applied
Yanping Chen, Xiuxiu Lin, Yunqing Huang
Summary: This paper investigates the spectral discretization of an optimal control problem governed by the space-time fractional diffusion equation with integral control and state constraints. The study focuses on the optimality conditions and introduces auxiliary equations and important operators to rigorously obtain a priori error estimates of spectral approximation for the state, adjoint state, and control variable. Additionally, reliable a posteriori error estimates are also provided for the optimal control problem. The analysis results indicate that the errors decay exponentially when the data is smooth.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Nitin Kumar, Mani Mehra
Summary: This paper presents a new method based on shifted Legendre polynomials for solving a class of two dimension fractional optimal control problem. The necessary optimality conditions are derived as a two-point fractional-order boundary value problem using the Lagrange multiplier method and integration by part formula. The fractional operators of shifted Legendre polynomial are computed and used to convert the necessary optimality conditions into a system of algebraic equations. L-2-error estimates in approximation and convergence analysis of the proposed method are discussed and illustrated with examples.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2022)
Article
Mathematics, Applied
Zuliang Lu, Fei Cai, Ruixiang Xu, Chunjuan Hou, Xiankui Wu, Yin Yang
Summary: In this paper, the spectral element approximation for the optimal control problem of parabolic equation is investigated, and an hp spectral element approximation scheme for the parabolic optimal control problem is presented. By using the Scott-Zhang type quasi-interpolation operator, a posteriori error estimates for both the state variables and the control variable are obtained. Additionally, posteriori error estimates for the hp spectral element approximation of the optimal parabolic control problem are derived by adopting two auxiliary equations and stability results.
Article
Automation & Control Systems
Sheril Kunhippurayil, Matthew W. Harris, Olli Jansson
Summary: This paper presents new mathematical results for lossless convexification of optimal control problems with a non-convex annular control constraint. The study focuses on a representative rocket landing problem and shows that controllability is a sufficient condition to solve fixed time problems as single convex programs. Controllability is also identified as a sufficient condition for solving the general fixed time problem as a sequence of convex programs.
Article
Mathematics, Applied
Alejandro Allendes, Francisco Fuica, Enrique Otarola, Daniel Quero
Summary: In Lipschitz polytopal domains in two and three dimensions, a reliable and efficient a posteriori error estimator for a semilinear optimal control problem with control constraints is devised and analyzed. The error estimator is decomposed into three contributions related to the discretization of state and adjoint equations, as well as the control variable. Results are extended to schemes approximating the control variable with piecewise linear functions and solutions to nondifferentiable optimal control problems, illustrated with numerical examples in two and three dimensions.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Jingrui Sun
Summary: This paper examines a linear quadratic optimal control problem with fixed terminal states and integral quadratic constraints. By introducing a Riccati equation and utilizing results from duality theory, the optimal control is derived as a target-dependent feedback of the current state.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Christian Glusa, Enrique Otarola
Summary: This paper translates the integral definition of the fractional Laplacian and linear-quadratic optimal control problem for the so-called fractional heat equation, including control constraints. By deriving existence, uniqueness results, and first order optimality conditions, a fully discrete scheme is proposed along with a new error estimation method. Finally, the theory is illustrated through one- and two-dimensional numerical experiments.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Automation & Control Systems
Jianping Huang, Jiongmin Yong, Hua-Cheng Zhou
Summary: A linear control system with a quadratic cost functional over infinite time horizon is considered without any assumptions. The existence and nonexistence of overtaking optimal controls in different cases are established, with concrete examples provided. These results demonstrate the potential and limitations of the overtaking optimality approach in solving related problems.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Francisco Fuica, Enrique Otarola, Daniel Quero
Summary: This work aims to derive a priori error estimates for finite element discretizations of control-constrained optimal control problems involving the Stokes system and Dirac measures. It discusses two problems involving reduced regularity properties in the solutions of state equations. Finite element solution techniques and numerical experiments are presented to illustrate the theoretical developments.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Xiaoliang Song, Bo Chen, Bo Yu
Summary: This paper addresses optimization problems with L-1-control cost functional subject to an elliptic partial differential equation. A new discretized scheme for the L-1-norm is proposed, showing advantages in terms of order of approximation compared to common nodal quadrature formula methods. Finite element error estimates results for the new scheme confirm that the approximation will not alter the order of error estimates. A symmetric Gauss-Seidel based majorized accelerated block coordinate descent method is introduced to solve the new discretized problem efficiently.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2021)
Article
Automation & Control Systems
Qi Xin, Rui Fu, Satish V. Ukkusuri
Summary: This study proposes a safe and sub-optimal longitudinal control protocol for CAV platoon with uncertain vehicle dynamics and state constraints. By encoding state constraints and speed trajectory tracking stability condition into control constraints, the control stability and performance of the CAV platoon are improved.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Martin Neumueller, Olaf Steinbach
Summary: This study analyzes the regularization error of state and target functions with respect to the regularization parameter in tracking type distributed optimal control problems with second-order elliptic partial differential equations. The focus is on regularization in the energy space H-1(omega), with higher-order convergence observed in the relaxation parameter for controls in this space. This higher-order convergence also affects the approximation of the target function by the state function.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Ram Manohar, Rajen Kumar Sinha
Summary: In this study, a posteriori error estimates of the finite-element method for linear parabolic optimal control problems in a convex bounded polyhedral domain are derived. Variational discretization is used to approximate the state, co-state, and control variables, while temporal discretization is based on the backward Euler method. The error analysis relies on the elliptic reconstruction technique and heat kernel estimates, resulting in posterior error estimates for the state, co-state, and control variables in the L-infinity(0, T; L-infinity(Omega))-norm.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
