On the “bang-bang” control problem
Authors:
R. Bellman, I. Glicksberg and O. Gross
Journal:
Quart. Appl. Math. 14 (1956), 11-18
MSC:
Primary 34.0X
DOI:
https://doi.org/10.1090/qam/78516
MathSciNet review:
78516
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Let $S$ be a physical system whose state at any time is described by an $n$-dimensional vector $x\left ( t \right )$, where $x\left ( t \right )$ is determined by a linear differential equation $dz/dt = Az$, with $A$ a constant matrix. Application of external influences will yield an inhomogeneous equation, $dz/dt = Az + f$, where $f$, the “forcing term", represents the control. A problem of some importance in the theory of control circuits is that of choosing $f$ so as to reduce $z$ to 0 in minimum time. If $f$ is restricted to belong to the class of vectors whose $i$th components can assume only the values $\pm {b_i}$, the control is said to be of the “bang-bang” type.
D. W. Bushaw, Ph.D. Thesis, Department of Mathematics, Princeton University, 1952
J. P. LaSalle, Abstract 247t, Bull. Amer. Math. Soc. 60, 154 (1954)
D. McDonald, Nonlinear techniques for improving servo performance, Cook Research Laboratories, Bulletin S-2, Chicago, 1950
N. J. Rose, Theoretical aspects of limit control, Report No. 459, Experimental Towing Tank, Stevens Institute of Technology, November 1953
- Leon Alaoglu, Weak topologies of normed linear spaces, Ann. of Math. (2) 41 (1940), 252–267. MR 1455, DOI https://doi.org/10.2307/1968829
T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Ergebnisse der Mathematik 3, 1, Berlin 1934, New York 1948
D. W. Bushaw, Ph.D. Thesis, Department of Mathematics, Princeton University, 1952
J. P. LaSalle, Abstract 247t, Bull. Amer. Math. Soc. 60, 154 (1954)
D. McDonald, Nonlinear techniques for improving servo performance, Cook Research Laboratories, Bulletin S-2, Chicago, 1950
N. J. Rose, Theoretical aspects of limit control, Report No. 459, Experimental Towing Tank, Stevens Institute of Technology, November 1953
L. Alaoglu, Weak topologies of normed linear spaces, Ann. of Math. 41, 252-267 (1940)
T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Ergebnisse der Mathematik 3, 1, Berlin 1934, New York 1948
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
34.0X
Retrieve articles in all journals
with MSC:
34.0X
Additional Information
Article copyright:
© Copyright 1956
American Mathematical Society