A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation

In this paper, we study a time optimal internal control problem governed by the heat equation in $\Omega\times [0,\infty)$. In the problem, the target set $S$ is nonempty in $L^2(\Omega)$, the control set $U$ is closed, bounded and nonempty in $L^2(\Omega)$ and control functions are taken from the set \uad=\{u(\cdot, t): [0,\infty)\ra L^2(\Omega) {measurable}; u(\cdot, t)\in U, {a.e. in t} \}. We first establish a certain null controllability for the heat equation in $\Omega\times [0,T]$, with controls restricted to a product set of an open nonempty subset in $\Omega$ and a subset of positive measure in the interval $[0,T]$. Based on this, we prove that each optimal control $u^*(\cdot, t)$ of the problem satisfies necessarily the bang-bang property: u^*(\cdot, t)\in \p U for almost all $t\in [0, T^*]$, where \p U denotes the boundary of the set $U$ and $T^*$ is the optimal time. We also obtain the uniqueness of the optimal control when the target set $S$ is convex and the control set $U$ is a closed ball.Comment: 22 page

Bang-bang property of time optimal controls for some semilinear heat equation

In this paper, we derive a bang-bang property of a kind of time optimal control problem for some semilinear heat equation on bounded $C^2$ domains (of the Euclidean space), with homogeneous Dirichlet boundary condition and controls distributed on an open and non-empty subset of the domain where the equation evolves

On the switching behavior of sparse optimal controls for the one-dimensional heat equation

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $\nu$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $\nu \ge 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $\nu \searrow 0$

A Game Problem for Heat Equation

In this paper, we consider a two-person game problem governed by a linear heat equation. The existence of Nash equilibrium for this problem is considered. Moreover, the bang-bang property of Nash equilibrium is discussed.Comment: 10 page

Time-varying Bang-bang Property of Minimal Controls for Approximately Null-controllable Heat Equations

In this paper, optimal time control problems and optimal target control problems are studied for the approximately null-controllable heat equations. Compared with the existed results on these problems, the boundary of control variables are not constants but time varying functions. The time-varying bang-bang property for optimal time control problem, and an equivalence theorem for optimal control problem and optimal target problem are obtained.Comment: 13 page

Two equivalence theorems of different kinds of optimal control problems for Schr\"{o}dinger equations

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schr\"{o}dinger equations. The first one concerns with the equivalence of the minimal norm and the minimal time control problems. (The minimal time control problems are also called the first type of optimal time control problems.) The targets of the aforementioned two kinds of problems are the origin of the state space. The second one deals with the equivalence of three optimal control problems which are optimal target control problems, optimal norm control problems and the second type of optimal time control problems. These two theorems were estabilished for heat equations in [18] and [17] respectively.Comment: 27 page

Observability inequalities from measurable sets for some evolution equations

In this paper, we build up two observability inequalities from measurable sets in time for some evolution equations in Hilbert spaces from two different settings. The equation reads: $u'=Au,\; t>0$, and the observation operator is denoted by $B$. In the first setting, we assume that $A$ generates an analytic semigroup, $B$ is an admissible observation operator for this semigroup (cf. \cite{TG}), and the pair $(A,B)$ verifies some observability inequality from time intervals. With the help of the propagation estimate of analytic functions (cf. \cite{V}) and a telescoping series method provided in the current paper, we establish an observability inequality from measurable sets in time. In the second setting, we suppose that $A$ generates a $C_0$ semigroup, $B$ is a linear and bounded operator, and the pair $(A, B)$ verifies some spectral-like condition. With the aid of methods developed in \cite{AEWZ} and \cite{PW2} respectively, we first obtain an interpolation inequality at one time, and then derive an observability inequality from measurable sets in time. These two observability inequalities are applied to get the bang-bang property for some time optimal control problems.Comment: 29 page

Approximation of time optimal controls for heat equations with perturbations in the system potential

In this paper, we study a certain approximation property for a time optimal control problem of the heat equation with $L^\infty$-potential. We prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation, where the potential has a small perturbation, are close to those for the original problem. We also verify that for the heat equation with a small perturbation in the potential, one can construct a new time optimal control problem, which has the same target as that of the original problem, but has a different control constraint bound from that of the original problem, such that the new and the original problems share the same optimal time, and meanwhile the optimal control of the new problem is close to that of the original one. The main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration. This theorem was first established by G.Wang and E. Zuazua in [20] for the case where the controlled system is an internally controlled heat equation without the potential and the target is the origin of the state space.Comment: 28 page

Optimal Shortcuts to Adiabaticity for a Quantum Piston

In this paper we use optimal control to design minimum-time adiabatic-like paths for the expansion of a quantum piston. Under realistic experimental constraints, we calculate the minimum expansion time and compare it with that obtained from a state of the art inverse engineering method. We use this result to rederive the known upper bound for the cooling rate of a refrigerator, which provides a quantitative description for the unattainability of absolute zero, the third law of thermodynamics. We finally point out the relation of the present work to the fast adiabatic-like expansion of an accordion optical lattice, a system which can be used to magnify the initial quantum state (quantum microscope)

Arbitrary n-Qubit State Transfer Implemented by Coherent Control and Simplest Switchable Local Noise

We study reachable sets of open n-qubit quantum systems, whose coherent parts are under full unitary control, by adding as a further degree of incoherent control switchable Markovian noise on a single qubit. In particular, adding bang-bang control of amplitude damping noise (non-unital) allows the dynamic system to act transitively on the entire set of density operators. Thus one can transform any initial quantum state into any desired target state. Adding switchable bit-flip noise (unital) instead suffices to get all states majorised by the initial state. Our open-loop optimal control package DYNAMO is extended by incoherent control to exploit these unprecedented reachable sets in experiments. We propose implementation by a GMon, a superconducting device with fast tunable coupling to an open transmission line, and illustrate how open-loop control with noise switching achieves all state transfers without measurement-based closed-loop feedback and resettable ancilla.Comment: 28 pages, 7 figures. Supersedes arXiv:1206.494