We consider a Mayer problem of optimal control, whose dynamic constraint is given by a convex-valued differential inclusion. Both state and endpoint constraints are involved. We prove necessary conditions incorporating the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition. Our results weaken the hypotheses and strengthen the conclusions of earlier works. Their main focus is to allow the admissible velocity sets to be unbounded, provided they satisfy a certain continuity hypothesis. They also sharpen the assertion of the Euler-Lagrange inclusion by replacing Clarke's subgradient of the essential Lagrangian with a subset formed by partial convexification of limiting subgradients. In cases where the velocity sets are compact, the traditional Lipschitz condition implies the continuity hypothesis mentioned above, the assumption of "integrable boundedness" is shown to be superfluous, and our refinement of the Euler-Lagrange inclusion remains a strict improvement on previous forms of this condition.

Advanced model‐based controllers are well established in process industries. However, such controllers require regular maintenance to maintain acceptable performance. It is a common practice to monitor controller performance continuously and to initiate a remedial model re‐identification procedure in the event of performance degradation. Such procedures are typically complicated and resource intensive, and they often cause costly interruptions to normal operations. In this article, we exploit recent developments in reinforcement learning and deep learning to develop a novel adaptive, model‐free controller for general discrete‐time processes. The deep reinforcement learning (DRL) controller we propose is a data‐based controller that learns the control policy in real time by merely interacting with the process. The effectiveness and benefits of the DRL controller are demonstrated through many simulations.

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The conventional and optimization based controllers have been used in process industries for more than two decades. The application of such controllers on complex systems could be computationally demanding and may require estimation of hidden states. They also require constant tuning, development of a mathematical model (first principle or empirical), design of control law which are tedious. Moreover, they are not adaptive in nature. On the other hand, in the recent years, there has been significant progress in the fields of computer vision and natural language processing that followed the success of deep learning. Human level control has been attained in games and physical tasks by combining deep learning with reinforcement learning. They were also able to learn the complex go game which has states more than number of atoms in the universe. Self-Driving cars, machine translation, speech recognition etc started to gain advantage of these powerful models. The approach to all of them involved problem formulation as a learning problem.

Abstract. We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type x(t) € X(t) for all t ; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.