Abstract
ABSTRACTOrdinary differential equations (ODEs) have a wide range of potential applications in science and engineering with regard to nonlinear dynamic systems. Frequently, there is a focus upon locating unique solutions to ODEs, with non-unique solutions being viewed as potentially problematic. However, some have recognized the importance of examining the character of non-unique solutions as well in order to properly understand the behaviour of physical systems. In some areas of engineering, notably control theory, the latter concern has become pressing. In this paper, by studying the asymptotic stability of second-order ordinary differential systems, we present a theorem creatively and prove it strictly by two lemmas. Using the criteria for non-unique solutions of first-order ordinary differential equations at points of equilibrium, we can solve engineering problems effectively. The applicability of this novel approach to the solution of engineering problems is provided through an example relating to the optimization of finite time controllers.
Highlights
Ordinary differential equations (ODEs) have many important applications across fields as diverse as automatic control (Golnaraghi & Kuo, 2010), the design of electrical systems (Bellen, Guglielmi, & Ruehli, 1999), the calculation of trajectories (Betts, 1998), the stability of aircraft (Cook, 2012), missile flight-planning (Kuo, Soetanto, & Chiou, 2015), and chemical reaction processes (Gillespie, 1976)
In the field of control theory, there has been a burgeoning interest in finite time stability, and within this domain, the prospective properties of non-unique solutions at equilibrium points (Bhat & Bernstein, 2002; Coron, 2006; Hong, Huang, & Xu, 1999). In relation to this interest, this paper presents a new way of arriving at the sufficiency conditions for a specific kind of ODEs with non-unique solutions
We looked at an initial candidate ODE and an associated second-order system and formulated four hypotheses upon which we could found an investigation of non-uniqueness
Summary
Ordinary differential equations (ODEs) have many important applications across fields as diverse as automatic control (Golnaraghi & Kuo, 2010), the design of electrical systems (Bellen, Guglielmi, & Ruehli, 1999), the calculation of trajectories (Betts, 1998), the stability of aircraft (Cook, 2012), missile flight-planning (Kuo, Soetanto, & Chiou, 2015), and chemical reaction processes (Gillespie, 1976). Some researchers have made a compelling case that is essential to understand the characteristics of non-unique solutions to ODEs because, without this understanding, it is not possible to fully model the behaviour of physical and dynamic systems (Gifford & Tomlinson, 1989; Yodovich, 2004) Despite this recognition of their potential importance, theoretical understanding remains limited. In the field of control theory, there has been a burgeoning interest in finite time stability, and within this domain, the prospective properties of non-unique solutions at equilibrium points (i.e. the origin) (Bhat & Bernstein, 2002; Coron, 2006; Hong, Huang, & Xu, 1999) In relation to this interest, this paper presents a new way of arriving at the sufficiency conditions for a specific kind of ODEs with non-unique solutions.
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