In this paper, we study the optimal time problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter. To begin with, we build an exact solution. The dependence of this solution as regards the scaling parameter naturally opens the way to study the existence and uniqueness of an optimal time control. If, moreover, one assumes the L∞− null controllability, it enab...

This paper presents a general necessary condition for singular time optimal control of robotic manipulation moving along specified paths. Early work [l], [7] ignored the issue of singular control, assuming bang-bang acceleration along the path. Recent work [6] has shown that the time optimal control can be singular if one of the equations of motion reduces to a velocity constraint. This paper d...

Abstract. This paper presents a new observability estimate for parabolic equations in × (0, T ), where  is a convex domain. The observation region is restricted over a product set of an open nonempty subset of  and a subset of positive measure in (0, T ). This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property f...

We provide sufficient optimality conditions for optimal control problems with bang-bang controls. Building on a structural assumption on the adjoint state, we additionally need a weak second-order condition. This second-order condition is formulated with functions from an extended critical cone, and it is equivalent to a formulation posed on measures supported on the set where the adjoint state...

We investigate two well-known basic optimal control problems for chemotherapeutic cancer treatment modified by introducing a timedependent “resistance factor”. This factor should be responsible for the effect of the drug resistance of tumor cells on the dynamical growth for the tumor. Both optimal control problems have common pointwise but different integral constraints on the control. We show ...

It is well known that the sufficient family of time-optimal paths for both Dubins’ as well as Reeds-Shepp’s car models consist of the concatenation of circular arcs with maximum curvature and straight line segments, all tangentially connected. These time-optimal solutions suffer from some drawbacks. Their discontinuous curvature profile, together with the wear and impairment on the control equi...

This paper presents a new observability estimate for parabolic equations in Ω × (0, T ), where Ω is a convex domain. The observation region is restricted over a product set of an open nonempty subset of Ω and a subset of positive measure in (0, T ). This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm a...

Abstract. This paper presents a new observability estimate for parabolic equations in × (0, T ), where  is a convex domain. The observation region is restricted over a product set of an open nonempty subset of  and a subset of positive measure in (0, T ). This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property f...

The distance from the convex hull of the range of an n-dimensional vector-valued measure to the range of that measure is no more than an/2, where a is the largest (one-dimensional) mass of the atoms of the measure. The case a = 0 yields Lyapounov's Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal control theory.

The effect of a zero in the right-half plane on the controller derived through the stability theorem of Lyapunov is considered through a simple illustrative example. It is shown that the controller should have the form of a linear saturating function instead of the bang-bang form for the minimum-phase systems. Many industrial processes are known to exhibit a dead-time delay in their txansfer ch...