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In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.

In this chapter, we will study the bang-bang property of some time optimal control problems. The bang-bang property of such a problem says, in plain language, that any optimal control reaches the boundary of the corresponding control constraint set at almost every time. This property not only is mathematically interesting, but also has important applications. For instance, the uniqueness of time optimal control is an immediate consequence of the bang-bang property in certain cases.

This paper studies a time optimal control problem governed by a classical controlled system of nonlinear ordinary differential equations. Corresponding to certain controls, the solutions of the system have the behavior of blowup; namely, they blow up at a finite time. The goal for controlling the system is to minimize the blowup time. Therefore, the time optimal control problem studied in this paper holds its target set outside of its state space. From this perspective, it essentially differentiates from the most time optimal control problems in studies from past publications. The purpose of this study is to develop a new method for proving the existence of time optimal controls for such kinds of time optimal control problems through analyzing a typical problem. We also aim to create a new method for establishing the Pontryagin maximum principle of optimal controls. The main results derived from this paper can be extended to controlled systems, where nonlinear terms have a more general form, by the two methods mentioned above.

This paper deals with the determination of the conditions of optimality for a control problem related to a rain history identification from available moisture observations in the soil.

The purpose of this paper is to study an identification problem related to a specific nonlinear model describing the rainfall type infiltration into a porous medium in which saturation can be partially or totally reached after some time.

This work deals with maximum principle of optimal control governed by some fluid dynamic systems, which include three-dimensional Navier-Stokes equations. Some type of state constraint is involved.

In this paper, we study two subjects on internally controlled heat equations with time varying potentials: the attainable subspaces and the bang-bang property for some time optimal control problems. We present some equivalent characterizations on the attainable subspaces, and provide a sufficient conditions to ensure the bang-bang property. Both the above-mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations. It seems for us that the existing ways to derive the bang-bang property for heat equations with time-invariant potentials (see, for instance, [4],[7],[16],[26]) do not work for the case where the potentials are time-varying. We provide another way to approach it in the current paper., 33pages

This work deals with the necessary conditions of optimality for some optimal control problems governed by elliptic variational inequalities. Boundary control and state constrained problems are considered. The techniques used are based on those in Ref. 1 and a new penalty functional is defined in this paper.

This paper presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal norm control problems and optimal time control problems. Controlled systems in this study are internally controlled heat equations. With the aid of this theorem, we establish an optimal norm feedback law and build up two algorithms for optimal norms (together with optimal norm controls) and optimal time (along with optimal time controls), respectively., Comment: 34 pages