This paper considers a nonlinear constrained optimal control problem (NCOCP) originated from energy optimal trajectory planning of servomotor systems. Solving the exact optimal solution is challenging because of the nonlinear and switching cost function, and various constraints. This paper proposes a method to manage the switching cost function to establish a set of necessary conditions of an NCOCP. Specifically, a concept ”sub-trajectory ” is introduced to match multiple Hamiltonian due to switches in the cost function. Necessary conditions on the optimal trajectory is established as a union of conditions for all sub-trajectories and Weierstrass-Erdmann corner conditions between sub-trajectories. The set of feasible structures of optimal trajectories is further identified and represented by various state transition diagrams for the servomotor application. A decomposition-based shooting method is proposed to compute an optimal trajectory by solving multi-point boundary value problems. Simulations and experiments validate the effectiveness of the methodology and the energy saving benefit.

This paper presents a preliminary study of economical manufacturing from optimal control perspective. We begin with modeling twomanufacturing systems as linear and nonlinear constrained continues-time dynamics, respectively. Economical manufacturing, characterized by minimal economical cost, is formulated as optimal control problems with variations on the cost function and system dynamics. The optimal control problems are interesting and challenging to solve due to constraints, nonlinearity, battery components, and consideration of energy price profiles. Rigorous analysis establishes the characteristics of optimal solutions and/or properties of optimal control problems. Methods and techniques to solve these optimal control problems are discussed for further investigation.

Optimal control of switched systems is challenging due to the discrete nature of the switching control input. The embedding-based approach is one of the most effective approaches for addressing such a challenge. It tries to compute the optimal switching input by solving a corresponding relaxed optimal control problem with only continuous inputs, and then projecting the relaxed solution back to obtain the optimal switching solution of the original problem. This paper presents a novel idea that views the embedding-based approach as a change of topology over the optimization space, resulting in a general procedure to construct a switched optimal control algorithm with guaranteed convergence to a local optimizer. Our result provides a unified topology-based framework for the analysis and design of various embedding-based algorithms in solving the switched optimal control problem and includes many existing methods as special cases. I.

This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0–1 linear program, which can be solved by existing polynomial-time algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function. The performance and utility of the proposed algorithms are demonstrated with the ON/OFF control problems of interdependent refrigeration systems. I.