China
Africa
Explore chapters and articles related to this topic
Optimal Control Theory
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
The Hamiltonian is the operator corresponding to the total energy of the system in most cases. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation with the time-evolution of a system, it is very important in most formulations of quantum theory. On the other hand, the Hamiltonian in optimal control theory is distinct from its quantum mechanical definition. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian. This is known as Pontryagin’s minimum principle which states that a control u(t) is to be chosen so as to minimize the objective function J(u)=Ψ(x(T))+∫0TL(x,u,t)dt, $$ \begin{aligned}J(u)= \Psi (x(T))+\int _0^T L(x,u,t)\,dt,\end{aligned} $$
Miscellaneous
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Define the Hamiltonian H(t,x,u,λ)=λTb(t,x,u)-c(t,x,u) $$ H(t,~x,~u,~\lambda ) = \lambda ^{T} b(t,~x,~u) - c(t,~x,~u) $$
On the Principles of Quantum Control Theory
Published in Ning Xi, Mingjun Zhang, Guangyong Li, Modeling and Control for Micro/Nano Devices and Systems, 2017
Re-Bing Wu, Jing Zhang, Tzyh-Jong Tarn
A nanoscale system can be modeled as either classical or quantum, although the former is nothing but an approximation of the latter under proper physical conditions. Mathematically, the quantum description is quantized from a classical model by replacing physical observables with operators that satisfy certain commutation relationships. In particular, to characterize the dynamics of quantum systems, it is essential to know the Hamiltonian that involves a potential energy function. As shown in Figure 1.1, the value of potential energy of a classical system can be an arbitrary real number. However, in quantum systems, only eigenvalues of the operator corresponding to the Hamiltonian can be recognized by a classical observer; they are called the energy levels. When the potential is a well, the energy levels are usually discrete, and the number of levels can be infinite when the well is infinitely deep (Figures 1.1b,c). The discretization of energy values is where we get the terminology "quantum," but it should be clarified that quantum energy levels are not always discrete, because a continuous spectrum (see Figure 1.1a) can exist when the energy is beyond the top of the potential.
Normal ordered exponential approach to thermal properties and time-correlation functions: general theory and simple examples
Published in Molecular Physics, 2021
It would appear that this state of affairs has mainly historical origins. Much of the original work on thermal properties has been done based on a quantum field theoretical framework, extending time-dependent propagator or Green's function theory to thermal properties, using in essence a Wick rotation, in addition to further analysis. In this historical context, a perturbative diagrammatic starting point is the essential point of departure, and any resulting theory is naturally rooted in perturbative expansions and partial re-summations. The PhD work of one of the authors took exactly this point of view for purely electronic problems, starting from perturbative diagrammatic expansions and back-engineering to infinite order theories like Coupled Cluster theory and the Coupled Cluster Green's function, using recursive diagrammatic procedures [29–31]. Interestingly, in this process, the Green's function was obtained directly, and not through an approximation to the irreducible self-energy and a subsequent solution of Dyson's equation. Moreover, this approach is basically equivalent to Coupled Cluster linear response theory, e.g. [32] or equation of motion CC theory [33]. This suggests that the methods of quantum field theory, which are defined from the outset in terms of a perturbation expansion may benefit from a reformulation when a closed form Hamiltonian is known. In the context of quantum chemistry, the picture is usually reversed. One has easy direct ways to define Coupled Cluster theory, as an approximation to full CI, and a perturbative expansion of the theory leads to many-body perturbation theory, in a manifestly connected form, e.g. [34–36].