Abstract
We are concerned here with the linear control system
where A is a (possibly unbounded) operator with domain D(A) in the Banach space E and range in E, B is a bounded operator from another complex Banach space F to E. As usual, we assume A to be the infinitesimal generator of a strongly continuous semigroup T(t), t > 0 ([4], Chapter VIII). If f( ∙) (the input or control) is any locally summable F-valued function and u is any element of E we define the “variation-of-constants” expression
to be a solution of (1.1), where the integral-on the right-hand side of (1.2) is a Bochner integral ([8], Chapter III). The solution (or trajectory or output) u( ∙) of (1.1) is always continuous and takes the value u for t = 0; if u, f( ∙) are “smooth” in one sense or another then u( ∙) is a genuine solution, i.e., it is continuously differentiable and satisfies (1.1) everywhere, while it is only a generalized solution in the general case. Solutions of (1.1) with given input and initial condition are unique. (See [2] for further discussion on these questions.)
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References
H. A. ANTOSIEWICZ, Linear control systems, Arch. Rat. Mech. Anal. 12 (1963) pp. 313–324.
A. V. BALAKRISHNAN, Optimal control problems in Banach spaces, J. Siam Control 3 (1965) pp. 152–180.
R. CONTI, On some aspects of linear control theory, Proceedings of the Conference on the Mathematical Theory of Control held in Los Angeles, California, Academic Press, New York, 1967, pp. 285–300.
N. DDNFORD - J. T. SCHWARTZ, Linear operators, part I, Interscience, New York, 1957.
N. DDNFORD, Linear operators, part II, Interscience, New York, 1963.
H. O. FATTORINI, Some remarks on complete controllability, J. SIAM Control 4 (1966) pp. 686–693.
H. O. FATTORINI, On complete controllability of linear systems, Journal Diff. Equations 3 (1967) pp. 391–402.
E. HILLE - R. S. PHILLIPS, Functional analysis and semigroups, Amer. Math. Soc., Providence, R. I., 1957.
K. HOFFMAN, Banach spaces of analytic functions, Prentice-Hall, Inc., Inglewood Cliffs, New Jersey, 1962.
R. E. KALMAN–T. C. Ho–K. S. NARENDRA, Controllability of linear dynamical systems, Contributions Diff. Equations 1 (1963) pp. 190–213.
R. E. KALMAN, Mathematical description of linear dynamical systems, J. SIAM Control 1 (1963) pp. 152–192.
W. MIRANKER, Approximate controllability for distributed linear systems, Journal Math. Anal. Appl. 10 (1965) pp. 378–387.
R. S. PHILLIPS, On weakly compact subsets of a Banach space, Amer. Journal Math. LXV (1943) pp. 108–136.
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Fattorini, H.O. (1969). Control with Bounded Inputs. In: Computing Methods in Optimization Problems. Lecture Notes in Operations Research and Mathematical Economics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85974-8_9
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DOI: https://doi.org/10.1007/978-3-642-85974-8_9
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