Probabilistic dynamic systems are appplied in technology, transportation, economics, medicine, electronic democracy etc. They are able to model complex technologies, lumphatic systems or group of automata known as one-arm bandits. Often, structure of the model is known but its parameters are unknown and has to be learnt. Often it has to be done jointly with influencing the system, with control. This creates an interesting and difficult problem as the chosen decisions (inputs, actions) influence both the system and learning efficiency. Reaching of a feasible balance between the corresponding expoitation and exploration is an interesting long standing problem that can and should be addressed with the novel fully probabilistic design of decision strategies.

# Topic for students

Fully probabilstic design of dynamic decision startegies is a well-developed theoretical basis of learning decision systems, which are potentially widely applicable in technology, natural and societal systems. The applicability is strongly constrained by complexity of the associated optimisation, a special version of dynamic programming. In the considered case, it is necessary to approximate a scalar function of many variables, which is implictily described as a solution of of non-linear integral-difference equation. The solution of this hard problem can be split into topics of several Phd theses, which will differ in the stress on functional analysis, approximation of functions or various heuristic methods encountered in connection with artificial intelligence. Also, software or simulation oriented solutions are welcome.

Hierarchical control and decision making connected with complex processes always contains a layer in which, decsions are made by human being, by an "operator". The proposed topic is related to a group of projects, which aims to create an advanced computer support of such decsion making. Existing original probabilistic theory already proved to efficient for this task. There is however a range of theoretical, algorithmic and software problems that remain to be solved in order to get widely applicable tool. This provides an interesting and useful area for research of 2-3 PhD students.