Minimum Time Function Research Articles (Page 1)

This paper develops a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove Hölder continuity of the minimum time function. We give explicit pointwise conditions of any order by using higher order hamiltonians which combine derivatives of the controlled vector field and the functions that locally define the target. For the controllability of a point our sufficient conditions extend some classically known results for symmetric or control affine systems, using the Lie algebra instead, but for targets of higher dimension our approach and results are new. We find our sufficient higher order conditions explicit and easy to compute for targets with curvature and general control systems. Some cases of nonsmooth targets are also included.

In this paper, a general formula concerning the multi-composition rule of convex subdifferential calculus is provided in the setting of Banach spaces under an appropriate regularity condition. As an application, this calculus rule is applied to obtain necessary and sufficient Karush–Kuhn–Tucker type optimality conditions for constrained convex minmax location problems with perturbed minimal time functions and set-up costs.

The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first provide a directional invariant point theorem. As direct consequences, we obtain several directional variational principles (with quite different formulations). Then, applying these results, we establish sufficient conditions for the existence of solutions for two general models of directional variational relation and inclusion problems. We also include corresponding consequences for particular cases.

In a normed space, we study the perturbed minimal time function determined by a bounded closed convex set and a proper lower semicontinuous function. It covers the perturbed distance function as a special case. In particular, we show that the ε-subdifferential and the limiting subdifferential of the perturbed minimal time function are representable by virtue of corresponding subdifferential of the associated function and the support function of the constraint set.

"For a class of semilinear control systems we get null controllability results and estimates for the minimum time function by considering appropriate feedback laws."

SUMMARY The reliability of the ground motion simulation is highly dependent on the quality of the site response (or site effects) evaluation. There are several methods to estimate the Site Amplification Factor (SAF) corresponding to Fourier Amplitude Spectra (FAS), either by using numerical simulation tools or empirical approaches. A widely used empirical method is the ‘Standard Spectral Ratio’ (SSR) technique based on the FAS ratio of the seismic record at a target site to the corresponding one at a nearby amplification-free ‘rock’ site (reference site). The main limitation of this method lies in the availability of a reference site relatively close to the target one. In this study a SAF estimation technique at a target site in relation to a distant reference site is presented and evaluated. This technique is based on the retrieval of the minimum phase Source Time Function (mpSTF) at a pair of examined sites (target-reference), with the Spectral Factorization analysis of Coda waves (SFC) proposed by Sèbe et al. The so derived mpSTF are considered as a convolution of the actual source function, and of the SAF, so that the FAS ratio of the mpSTF, derived at one site (target) and at a distant reference site, should be an estimate of the target SAF. The latter is confirmed in this study. Under the conditions of a common STF at the examined sites and of similar coda waves excitation factor, the ratio of the FAS of the mpSTFs (target over reference site) can safely approach the actual SAF, at least when target-reference distance is up to ∼60 km and provides encouraging results at longer distances. This technique was applied at 24 sites in western Greece in relation to 4 reference sites located at varying distances from the target ones (from 0.4 to 110 km). More than 700 STFs were calculated for 89 moderate magnitude earthquakes (3.9 ≤ M ≤ 5.1) located in this high seismicity area and SAFs were determined from each pair of target-reference stations using common seismic sources. Finally, the average SAFs were computed and compared to the ones computed by other methods (e.g. SSR, GIT and HVSR) demonstrating the reliability and robustness of the proposed technique in site effect estimation.

The key purpose of this project is to invent a mechanism for high/low voltage tripping that will protect against danger. Microcontroller-based relays run rapidly and have less running time. Frequent variations are frequent in the availability of AC mains and the business. A low path of resistance contributes to an extra current flow. Due to multiple causes, irregular overvoltages can occur. To secure the load, the tripping function is nice to have. The purpose of our project is to secure electrical devices using an Arduino above and below the voltage. The primary aim of this relay is to isolate the load by monitoring the relay tripping coil with Arduino from the over and under-voltage conditions. It detects any voltage greater than or less than 230V AC (pre-set value). In turn, when the voltage is greater than / less than the current value, it begins a trip signal that is provided to the circuit breaker, then removes the load from the circuit breaker source. Also, for relays, the Definite Minimum Time (IDMT) function is used. It is possible to expand this project further to over content relays. As the sum of supply voltage increases, the trip time decreases for reverse time features and the relay operates accordingly. The relay operates on the stated time characteristics only after 5 seconds of failure, regardless of the sum of device voltage. Where the voltage returns to the pre-set value, the relay is reset during relay service (230V AC).

In this paper, we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of the closedness of target sets with respect to constant dynamics. Then, we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.

In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called \emph{multiagent system} made of several possible interacting agents.

In this paper, we propose and study a new notion of local error bounds for a convex inequalities system defined in terms of a minimal time function. This notion is called generalized local error bounds with respect to F, where F is a closed convex subset of the Euclidean space $${\mathbb {R}}^n$$ satisfying $$0\in F$$ . It is worth emphasizing that if F is a spherical sector with the apex at the origin then this notion becomes a new type of directional error bounds which is closely related to several directional regularity concepts in Durea et al. (SIAM J Optim 27:1204–1229, 2017), Gfrerer (Set Valued Var Anal 21:151–176, 2013), Ngai and Thera (Math Oper Res 40:969–991, 2015) and Ngai et al. (J Convex Anal 24:417–457, 2017). Furthermore, if F is the closed unit ball in $${\mathbb {R}}^n$$ then the notion of generalized local error bounds with respect to F reduces to the concept of usual local error bounds. In more detail, firstly we establish several necessary conditions for the existence of these generalized local error bounds. Secondly, we show that these necessary conditions become sufficient conditions under various stronger conditions of F. Finally, we state and prove a generalized-invariant-point theorem and then use the obtained result to derive another sufficient condition for the existence of generalized local error bounds with respect to F.

Lipschitz regularity of the minimum time function of differential inclusions with state constraints

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Holder regularity of the minimum time function in optimal control. We then apply such result to prove Holder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the Holder regularity are essentially necessary, at least for smooth vector fields and target.

Minimum time and minimum energy for linear control systems

In this paper, we first present formulas for computing the Frechet subdifferentials and Frechet singular subdifferentials of the minimal time function for a differential inclusion in R n with a general target K. These formulas are characterized in terms of Frechet normal cones to a sub-level set of and a level set of the minimized Hamiltonian associated with the differential inclusion. Our results extend, improve and complement to existing results. An application to sensitivity analysis for a class of time optimal control problems is also given.

We define the minimal time function associated with a collection of sets which is motivated by the optimal time problem for nonconvex constant dynamics. We first provide various basic properties of this new function: lower semicontinuity, principle of optimality, convexity, Lipschitz continuity, among others. We also compute and estimate proximal, Fréchet and limiting subdifferentials of the new function at points inside the target set as well as at points outside the target. An application to location problems is also given.

We define and study C1-solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that C1-solutions are absolutely minimizing functions. We discuss how C1-supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct C1-solutions (even classical) explicitly.

This paper is concerned with a kind of minimal time control problem for a linear evolution equation with impulse controls. Each problem depends on two parameters: the upper bound of the control constraint and the moment of impulse time. The purpose of such a problem is to find an optimal impulse control (among certain control constraint set), which steers the solution of the evolution equation from a given initial state to a given target set as soon as possible. In this paper, we study the existence of optimal control for this problem; by the geometric version of the Hahn–Banach theorem, we show the bang–bang property of optimal control, which leads to the uniqueness of the optimal control; we also establish the continuity of the minimal time function of this problem with respect to the above mentioned two parameters, and discuss the convergence of the optimal control when the two parameters converge.

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., if only Holder continuity holds) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Holder continuous or even discontinuous.

This paper presents the equivalence between minimal time and minimal norm control problems for internally controlled heat equations. The target is an arbitrarily fixed bounded, closed, and convex set with a nonempty interior in the state space. This study differs from [G. Wang and E. Zuazua, SIAM J. Control Optim., 50 (2012), pp. 2938--2958], where the target set is the origin in the state space. When the target set is the origin or a ball, centered at the origin, the minimal norm and the minimal time functions are continuous and strictly decreasing, and they are inverses of each other. However, when the target is located in other place of the state space, the minimal norm function may be no longer monotonous and the range of the minimal time function may not be connected. These cause the main difficulty in our study. We overcome this difficulty by borrowing some idea from the classical rising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on pp. 121--122 in [E. M. Stein and R. Shakarchi, Real Analy...