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Swinging control of nonlinear oscillations. (English) Zbl 0866.93049

The equation of a controlled plant is: \(dx/d=F(x,u,t)\), \(t\geq 0\), where \(u\) denotes the control. An equivalent Hamiltonian system is defined in terms of phase coordinates \(q\) and \(p\). \(\dot p=\partial H(p,q,u)/\partial q\), \(\dot q=-\partial H(p,q,u)/\partial p\). Observe that \(q=x\) is not necessarily true. The Hamiltonian \(H\) is linear in \(u\). Let \(H_0\) denote the Hamiltonian for the uncontrolled system, i.e. \(H_0=H(p,q,0)\). The energy of the uncontrolled system is denoted by \(S=\{(p,q): H=H_0\}\). Because of linear dependence on \(u\), we have: \(H(p,q,u)=H_0+H_1(p,q)u\). The objective demand is: \(H_0(p,q)\rightarrow H*\) as \(t\rightarrow \infty\). Let us regard the functional \(Q={1\over 2}[H_0(p,q)-H*]^2\) as the cost functional. A calculation shows that \(\dot Q=(H_0(p,q)-H*)[H_0,H_1]\), where \([,]\) denotes the Poisson bracket. To design the speed gradient algorithm, the author computes the speed \(\omega\) of change of \(Q\) along a trajectory of the system. Subject to assumptions on smoothness and convexity of \(\omega(x,u,t)\) he also proves boundedness of the cost function along any trajectory. If \(\omega(x,u,t)\) is bounded from above by a positive, continuous function, then \(\lim_{t\to \infty} \nabla_u\omega(x,u,t)=0\) is attained when \(u\) is the optimal control.
As an example of application, the author computes the optimal control of a simple two-dimensional pendulum.
Reviewer: V.Komkov (Roswell)

MSC:

93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
70H05 Hamilton’s equations
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References:

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