Solution blow-up is a well-known phenomenon occurring in many Ordinary Differential Equations (ODEs), including ODEs describing real physical systems. For both theoretical and practical (prevention of technogenic catastrophes) reasons, it is important to identify control strategies that dampen blow-up solutions.
First-order non-linear ODEs are used where the Right-Hand Side (RHS) of the equation remains convex and continuous, but otherwise, an arbitrary, non-linear function of unknown solution is considered. RHS also contains a parameter that generally may have different values in different physical circumstances.
A specific control objective (boundedness, at all times, of the solution by a specified threshold) is imposed. This control strategy is designed considering the system parameter as a control variable (practically, variation requires control of the system). The methodology for finding a suitable control consists of constructing a majorizing function. The majorizing function provides an upper bound for the RHS of the equation, and thus, due to the basic comparison theorem of ODEs, the solution of the ODE with the majorizing function on the RHS (the “majorized equation”) provides the bounds for the solution of the original ODE. The majorizing function, utilizing analytical solutions of a majorized equation with specific control variable profiles, is constructed.
A major result is then obtained by applying the comparison theorem of ODEs. This ensures that the solution of the original, un-majorized equation, with an identical control variable profile, also remains bounded over infinitely large time intervals and, moreover, below an a priori-imposed threshold.
Specific results are obtained and considered an example of the control of thermal explosion. Several explicit control variable profiles are considered, and analytical solutions for majorized equations are obtained for all of them. This provides estimations of the bounded behavior of the solutions of the original ODE under identical control conditions.
In conclusion, control strategies, preventing solution blow-up are constructed for a certain class of ODEs.
