That equation is called a (continuous-time) algebraic Riccati equation, and there is ample literature on when they are solvable; just look for this search term. For instance, the book Algebraic Riccati equations by Lancaster and Rodman, or Numerical solution of AREs by Bini, Iannazzo, Meini.
Federico Poloni
- 20.2k
- 2
- 82
- 120