Calculus of variations and optimal control; optimization Optimality conditions in control Example

Fuller's phenomenon

Builds on Chattering control Bang–bang control Pontryagin's maximum principle

Example

Fuller's problem is the optimal control problem $\text{minimize } \int_0^\infty x_1(t)^2 \, dt \quad \text{subject to } x_1' = x_2, \; x_2' = u, \; \abs{u(t)} \le 1, \; x(0) = \xi,$ where $\xi \in \R^2$ is a given nonzero initial state and $u$ is a measurable control. For every $\xi \ne 0$ the optimal control is bang–bang with $u = \pm 1$ . It switches infinitely many times, and the switching times accumulate at a finite time $t^*$ , at which the state reaches the origin. After $t^*$ the optimal control is $u = 0$ and the state stays at the origin. So the optimal control is a chattering control. This behaviour is called Fuller's phenomenon.

Remarks

A scaling argument explains the pattern. If $(x_1(t), x_2(t), u(t))$ is admissible, then so is $(\lambda^2 x_1(t/\lambda), \lambda x_2(t/\lambda), u(t/\lambda))$ for every $\lambda > 0$ , and its cost is multiplied by $\lambda^5$ . So the optimal feedback is invariant under this scaling, and the time between two switches shrinks by a fixed factor at each switch. The intervals then form a geometric series with finite sum, which gives the finite time $t^*$ . The optimal feedback has the form $u = -\sgn(x_1 + C x_2 \abs{x_2})$ for a certain constant $C > 0$ . Necessary conditions of Pontryagin's maximum principle together with this scaling are the main tools of the analysis.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Absolute continuity concept
Bang–bang control concept
Chattering control concept
Control system concept
Pontryagin's maximum principle statement
Fuller's phenomenon example

Tag	`RSZS9SFFZ7`
URI	`m2://RSZS9SFFZ7`
Kind	example
Writing	W0 stub
MSC	49K15, 49J15, 93C15
Instance of	Chattering control
Uses	Chattering control, Bang–bang control, Pontryagin's maximum principle

Fuller's phenomenon

Example

Remarks

Reading path

Further reading

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