The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

Stabilization of the motions of non-autonomous mechanical systems

“…In the case of an even number of coordinates n = 2k, the unperturbed motion (1.2) of system (2.1) will be stabilized up to uniform asymptotic stability regardless of the form of the non-linear forces by adding dissipative forces with diagonal matrix b(t)E and gyroscopic forces with matrix G(t), such that Remark 2. Theorem 2 and Corollary 2 supplement Theorem 3.2 in Kosov's paper 6 in that no constraints are imposed on the derivativeṖ(t) of the matrix of the non-conservative positional forces and the coefficient b(t) in front of the matrix of the dissipative forces can depend on time.…”

“…Note that such a problem was previously considered in Ref. 6, where the following condition for exponential stability was obtained (3.25) It is notable that the first inequality in condition (3.25) follows from inequality (3.24). Unlike the second inequality in (3.25), no upper limit is imposed on the derivativek(t) in condition (3.24).…”

“…7-9 Several theorems regarding the exponential stability and stabilization of the motions of non-autonomous mechanical systems with non-conservative forces have also been obtained by using a Lyapunov function of quadratic form. 6 An approach based on constructing a vector Lyapunov function with components of the vector norm type and a reference system and using the theory of limit functions 10-12 is employed to solve the problem of stabilizing the motions of non-autonomous mechanical systems. Unlike the classical comparison method, 1 this approach enables the requirements improved on the reference system to be relaxed and thereby enables us to supplement the results of other investigators, obtained using a scalar Lyapunov function of quadratic form.…”

“…(3.2); therefore, it is impossible to apply exponential stability theorem 2.1 from Kosov's paper6 to system (3.2).…”

The stability and stabilization of non-linear, non-stationary mechanical systems

“…As shown previously, 19 for an even number of coordinates, a linear potential system with a constrained matrix of potential forces can be stabilized to exponential stability by adding dissipative, gyroscopic and non-conservative positional forces with constant matrix coefficients. Here, the chosen matrix of non-conservative positional forces should be fairly "large", and the given forces will be prevalent in the closed system.…”