# Mathematical problems of quantum control

Quantum control studied possibilities to manipulate atomic and molecular systems with quantum dynamics using an external control field, e.g., a shaped laser pulse. It is an important area of research which attracts high interest due to various existing and prospective applications in quantum technologies ranging from quantum information and computing to laser control of chemical reaction and photochemistry. Examples of applications include breaking a chemical bond by laser, manipulation by a qubit, etc. In each quantum control problem the goal is to find a shape of the control pulse which optimally achieves the control goal. The control goal can be mathematically represented as a functional of the control pulse. The control problem can be formulated as maximization of the objective functional. In this talk we will outline some problems in quantum control including analysis of complete controllability of open quantum systems and investigation of quantum control landscape which is the graph of the objective functional. Specific property of the quantum control landscape is the number of traps, i.e. local but not global optima. This property has pratical importance as determining the level of difficulty of finding globally optimal controls. Recent results in this field include proof of the absence of traps for state manipulatin and gate generation for a single qubit, absence of traps for control of quantum transmission, the existence of trapping behaviour for multilevel quantum systems with special symmetries, etc.