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Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. [1] It has numerous applications in science, engineering and operations research.
Widely regarded as a milestone in optimal control theory, the significance of the maximum principle lies in the fact that maximizing the Hamiltonian is much easier than the original infinite-dimensional control problem; rather than maximizing over a function space, the problem is converted to a pointwise optimization. [8]
The relationship between these two value functions is called the "Bellman equation". In this approach, the optimal policy in the last time period is specified in advance as a function of the state variable's value at that time, and the resulting optimal value of the objective function is thus expressed in terms of that value of the state variable.
Optimal control is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has ...
The field of stochastic control has developed greatly since the 1970s, particularly in its applications to finance. Robert Merton used stochastic control to study optimal portfolios of safe and risky assets. [7] His work and that of Black–Scholes changed the nature of the finance literature.
[10] [11] [12] Unscented optimal control is a special case of tychastic optimal control theory. [ 1 ] [ 5 ] [ 13 ] According to Aubin [ 13 ] and Ross, [ 1 ] tychastic processes differ from stochastic processes in that a tychastic process is conditionally deterministic.
Its solution is the value function of the optimal control problem which, once known, can be used to obtain the optimal control by taking the maximizer (or minimizer) of the Hamiltonian involved in the HJB equation. [2] [3] The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and ...
The associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different. [5] It is possible to compute the expected value of the cost function for the optimal gains, as well as any other set of stable gains. [12] The LQG controller is also used to control perturbed non-linear ...