Higher order sufficient conditions involving possibly nonconvex sets of variations (without Lagrange multipliers) are established for the following property: \(\Phi_ 1(.):\) \(Q\subset Y\to R^ m\) is said to be strongly locally (\({\mathcal U},\Phi_ 2,C)\)-controllable at \(\bar q\in Q\) if there exist neighbourhoods \(G_ 1,G_ 2\) of 0 in \(R^ m,Z\), respectively, such that \(\Phi_ 1(\bar q)+G_ 1\subset \{\Phi_ 1(u)\); \(u\in {\mathcal U}\), \(\Phi_ 2(u)+G_ 2\subset C\}\) where Y is a vector space, Z is a topological vector space, \(Q\subset Y\), \(C\subset Z\) are convex subsets, Int(C)\(\neq \emptyset\), \({\mathcal U}\subset Q\) and \(\Phi_ 2(.):\) \(Q\to Z.\) More specialized conditions involving Lagrange multipliers and generalizing existing results are also presented. In the case of optimization problems these conditions rule out candidates satisfying first-order necessary optimality conditions. An example showing that the non-Lagrangean higher order conditions are stronger than the corresponding Lagrangean conditions is presented.