Controllability of dynamical systems with constraints. (English) Zbl 1129.93326

Summary: The article is devoted to analysing the approximate controllability without constrains and constrained controllability with non-negative controls of a given type second order infinite dimensional system. The considered dynamical system is governed by evolution equation with three damping terms and three terms without derivatives. Following this aim spectral theory for linear unbounded operators is involved. At first, remind the representation of considered infinite dimensional dynamical system by the infinite series of finite dimensional systems. Next, two theorems on necessary and sufficient conditions of unconstrained and constrained approximate controllability of considered system are formulated and proved. Finally, proven theorems are applied to two particular dynamical systems, including one infinite dimensional.


93B05 Controllability
Full Text: DOI


[1] Brammer, R. F., Controllability in linear autonomous systems with positive controllers, SIAM Journal on Control and Optimization, 10, 339-353 (1972) · Zbl 0242.93007
[2] D. Burgree, N.Y Brooklyn, Free vibrations of pin-ended column with distance between pin ends, J. Applied Mechanics (1951) 135-139.; D. Burgree, N.Y Brooklyn, Free vibrations of pin-ended column with distance between pin ends, J. Applied Mechanics (1951) 135-139.
[3] Chen, C. T., Introduction to Linear System Theory (1970), Holt: Holt Rinehart and Winston, New York · Zbl 0212.36602
[4] Chen, G.; Russel, D. L., A mathematical model for linear elastic systems with structural damping, Quaterly of Applied Mathematics, 39, 433-454 (1982) · Zbl 0515.73033
[5] Fattorini, H. O., Some remarks on complete controllability, SIAM Journal on Control and Optimization, 4, 686-694 (1966) · Zbl 0168.34906
[6] Fattorini, H. O., On complete controllability of linear systems, J. Differential Equations, 3, 391-402 (1967) · Zbl 0155.15903
[7] Huang, F., On the mathematical model with analytic damping, SIAM J. Control Optimization, 26-3, 714-724 (1988) · Zbl 0644.93048
[8] Ito, K.; Kunimatsu, N., Stabilization of non-linear distributed parameter vibratory system, International Journal of Control, 48, 2389-2415 (1988) · Zbl 0663.93053
[9] Ito, K.; Kunimatsu, N., Semigroup model of structurally damped Timoshenko beam with boundary input, International Journal of Control, 54, 367-391 (1991) · Zbl 0735.35112
[10] Klamka, J., Controllability of Dynamical Systems (1991), Kluwer: Kluwer Dordrecht · Zbl 0732.93008
[11] Sakawa, Y., Controllability for Partial Differential Equations of Parabolic Type, SIAM Journal on Control and Optimization, 12, 389-400 (1974) · Zbl 0289.93010
[12] Sakawa, Y., Feedback control of second order evolution equations with damping, SIAM Journal Control and Optimisation, 22, 343-361 (1984) · Zbl 0551.58010
[13] Sakawa, Y., Feedback stabilization of linear diffusion system, SIAM J. Control and Optimization, 21, 5, 667-675 (1983) · Zbl 0527.93050
[14] Triggiani, R., Controllability and observability in banach space with bounded operators, SIAM Journal on Control and Optimization, 13, 462-491 (1975) · Zbl 0268.93007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.