Uncertain Optimal Control With Jump And Applications

The complexity of the world makes the non-deterministic events we face in various forms. Non-determinacy mainly includes objective non-determinacy and subjective non-determinacy. Randomness is an objective non-determinacy. The probability theory is the effective tool to study the non-determinacy. And subjective non-determinacy, which is called the fuzziness by some scholars, usually is studied by Zadeh’s fuzzy set theory. However, a lot of surveys showed that in the real life, a lot of non-determinacy are neither explained by probability theory nor by fuzzy set theory. For example, some information and knowledge, which usually are represented by human language like "about100km","high speed","roughly80kg","approximately39℃", and "moderate size", neither satisfy properties of the probability theory nor properties of fuzzy set theory. A new theory is needed to deal with this kind of subjective non-determinacy. In2007, Professor Liu Baoding of Tsinghua University proposed an uncertainty theory based on normality, self-duality, countable subadditivity and product measure axioms, which is a new tool for studying subjective non-deterministic phenomena. Now it has been developed as a branch of mathematics. Uncertain optimal control is a new type of optimal control problem based on the uncertainty theory. In2010, Professor Zhu Yuanguo introduced and dealt with an uncertain optimal control problem by using dynamic programming. However, the study on uncertain optimal control is just beginning. Many problems for uncertain optimal control need to be studied further. In real word, some uncertain systems may be disturbed by some sudden external extreme events or noises, and the state of a system may show a sudden jump. Based on studying of uncertain optimal control without jump and taking full account of the influence of jump on uncertain systems, this dissertation will study uncertain optimal control problems with jump by applying dynamic programming principle. The followings are the main research contents:1. In this dissertation, firstly, a Z-jump uncertain variable Z(r1,r2,t) is introduced, and the linear additive properties of independent Z-jump uncertain variables is proved. Then a V-jump uncertain process, which is used to describe the discontinuous jump part in an uncertain differential system is introduced, and an existence theorem of a V-jump uncertain process is verified. Next, an uncertain integral and differential of any uncertain process with respect to a V-jump uncertain process are defined. At last, an uncertain differential equation with jump is introduced and the existence and uniqueness theorem of solution for an uncertain differential equation with jump is established.2. A continuous time model for uncertain optimal control with jump in one-dimensional case is put forward, the principle of optimality by using dynamic programming is given, the equation of optimality for the proposed model is obtained and applications of the model in optimal portfolio selection and optimal pension funds control is discussed.3. As the extension of one-dimensional continuous time model, a multidimensional uncer-tain optimal control model is further presented, the principle of optimality and the equation of optimality of the model are obtained and an application of the model in R&D (Research and Development) fiscal subsidy policy is studied.4. A kind of special uncertain optimal control problem with jump:linear-quadratic (LQ) uncertain optimal control problem with jump in one-dimensional case is studied, a necessary and sufficient condition for the existence of optimal LQ control is given, and the presented model is employed to solve an enterprize’s investment decision problem.5. The special continuous time uncertain LQ optimal control model from one-dimensional case is extended to multidimensional case, a necessary and sufficient condition for the exis-tence of optimal LQ control in the multidimensional case is obtained, and a factory’s optimal inventory model is provided to illustrate the effectiveness of the results obtained.