Variation Analysis Of The Minimal Time Function

In the field of optimal control, the minimal time function is a class ofvalue function. It is an important topic to study properties of the minimal timefunction. In the field of nonsmooth analysis, the distant function is a special caseof the minimal time function, which plays a very important role in nonsmoothanalysis.This thesis analyzes di?erential properties of the minimal time function andits regularity. Generally, the minimal time function is nonsmooth. Therefore,we mainly discuss the properties of generalized di?erentials(also called subd-i?erentials). It is shown that subdi?erentials of a minimal time function arerepresentable by the corresponding normal cones and minimized Hamiltonianfunction.Firstly, we consider the minimal time function of di?erential inclusion withconstant mapping, that is, the set-valued mapping in the di?erential inclusiontakes a fixed set U as its values. In a normed vector space, we prove that theminimal time function is well-posed, show that proximal and Fr′echet subdi?er-entials of the minimal time function are representable by virtue of corresponding normal cones and level or suplevel sets of the support function of U, and givean application in regularity. Moreover, an example is presented to show that theboundedness of U is indispensable . Compared with those known results in theliterature: the set U is not symmetrical; the set U does not necessarily containthe origin as an interior point, actually U can have empty interior.Next, we consider the minimal time function with moving target set. Apply-ing the Ekeland variational principle, we prove that the minimal time functionis well-posed, obtain the characterization ofε? subdi?erentials of minimal timefunction, then show that the limiting subdi?erentials of minimal time functionare representable by the normal cones and level or suplevel sets of the supportfunction of U.Third, for the minimal time function determined by linear control system,we show that its proximal and Fr′echet subdi?erentials are representable by thenormal cones and the level or suplevel sets of the corresponding minimized Hamil-tonian function.Forth, in Bananch space, we consider the minimal time function of semi-linear control system. By the corresponding Yosida approximation equation, weobtain regularity of the minimal time function. Especially, we prove the rela-tionships between the minimal time functions of Yosida approximation equationand the minimal time function of semilinear control system, then show proximal subdi?erential of the minimal time function can be represented by the normalcones and the level or suplevel sets of the corresponding minimized Hamiltonianfunction.Finally, we consider the minimal time function of di?erential inclusion withnonconstant mapping. In Hilbert space, we prove the well-posed property of theminimal time function, the invariant property of its Fr′echet subdi?erential, andcharacterizations of proximal and Fr′echet subdi?erentials.