## 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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