High Order Necessary Conditions for Optimality

Quadratic order conditions for bang-singular extremals

“…Goh in [18] proposed a special change of variables obtained via a linear ODE and in [17] used this transformation to show a necessary condition for the vector control problem. An extensive survey of these articles can be found in Gabasov and Kirillova [16]. Jacobson and Speyer in [21], and together with Lele in [22] obtained necessary conditions by penalizing the cost functional.…”

“…Jacobson and Speyer in [21], and together with Lele in [22] obtained necessary conditions by penalizing the cost functional. Gabasov and Kirillova [16], Krener [26], Agrachev and Gamkrelidze [1] obtained a countable series of necessary conditions that in fact exploit the idea of Goh transformation. Milyutin in [31] discovered an abstract essence of this approach and obtained even stronger necessary conditions.…”

Optimal control of a production-inventory system with state constraints and a quadratic cost criterion

“…And we know that it is optimal if it is an admissible solution, because of (21) (see Gabasov, Kirillova [4]). …”

“…T the gradient of the singular trajectory is [ -2c%, 0] r , i. e. it is directed in an inadmissible région with respect to (4 Thus we have an implicit function where we seek for a % satisfying (3), a and Tfixed, a t s e(0, T).…”

“…To get an expression for u (t) one has to evaluate the total derivatives of <p(t) along the trajectories of (11) and set them equal to zero (Gabasov, Kiriilova [4] …”