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The unsteady partial differential equations (UPDE) with convection term gives a clear descriptions for the solidification process of a slab in dynamic production of continuous casting. To give a suitable setting value of secondary cooling water flow rate for the dynamic control system, this study investigates an optimal control problem (OCP) of UPDE with convection term. Firstly, control vector discretization of OCP and the solution of UPDE are given. Secondly, due to the rapidity for gradient, this paper analyzes the expression of the gradient calculation method based on Hamiltonian function costate system by approximate treatment, matrix calculation and composite trapezoidal integral method. Thirdly, an improved three-term spectrum conjugate gradient algorithm (ITSCGA) is proposed to solve the OCP of UPDE, and the global convergence of the ITSCGA is demonstrated. Lastly, the performance of ITSCGA is demonstrated by experimental simulations. The results demonstrate that the ITSCGA provides a smaller temperature fluctuations, and improves the quality of a slab. [ABSTRACT FROM AUTHOR]

This paper deals with the orbit tracking and formation motion problems with optimal energies and reduced computational cost. First, the orbit tracking and formation motion are decomposed into movements in the normal and tangent directions of level orbits, respectively, and simultaneously, the optimal value functions in both directions are defined. Then, to reduce the computational cost, a switching threshold event‐triggered (STET) mechanism is designed. Based on the STET mechanism, the optimal value functions are constructed to evaluate the optimal energies of orbit tracking and formation motion. Critic neural networks are then designed to approximate the optimal value functions, which yield the optimal policies along the normal and tangent directions of desired orbits, that is, a so‐called switching threshold event‐triggered critic algorithm (STET‐C). Theoretical analysis of system convergence is given in detail. Finally, two comparison simulations are given. The former intends to verify the optimal energy of STET‐C compared to the feedback controllers. The latter shows that STET‐C significantly reduces the computational cost in contrast with the non‐triggered actor‐critic algorithm, non‐triggered critic algorithm, and the relative threshold event‐triggered critic algorithm. [ABSTRACT FROM AUTHOR]

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O (1 k 2 ) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm. [ABSTRACT FROM AUTHOR]

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. [ABSTRACT FROM AUTHOR]

We consider the application of the Douglas-Rachford (DR) algorithm to solve linearquadratic (LQ) control problems with box constraints on the state and control variables. We have split the constraints of the optimal control problem into two sets: one involving the ordinary differential equation with boundary conditions, which is affine, and the other, a box. We have rewritten the LQ control problems as the minimization of the sum of two convex functions. We have found the proximal mappings of these functions, which we then employ for the projections in the DR iterations. We propose a numerical algorithm for computing the projection onto the affine set. We present a conjecture for finding the costates and the state constraint multipliers of the optimal control problem, which can, in turn, be used to verify the optimality conditions. We conducted numerical experiments with two constrained optimal control problems to illustrate the performance and the efficiency of the DR algorithm in comparison with the traditional approach of direct discretization. [ABSTRACT FROM AUTHOR]

This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment's polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b's bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm's main flaw is its long runtime due to the numerical approximation of the Jacobians. [ABSTRACT FROM AUTHOR]

This work focuses on an exploration of the pointwise Jacobson-type necessary conditions for optimal control problems governed by differential systems with impulse at fixed times; the pointwise Jacobson-type necessary optimality conditions refer to a type of pointwise second-order necessary optimality conditions for optimal singular control in the classical sense. By introducing an impulsive linear matrix Riccati differential equation, we derive the integral representation of the functional second-order variational. Based on this, the integral form of the second-order necessary conditions and the pointwise Jacobson-type necessary conditions are obtained. Incidentally, we have established the Legendre-Clebsch condition and the pointwise Legendre-Clebsch condition. Finally, an example is provided to illustrate the effectiveness of the main result. [ABSTRACT FROM AUTHOR]