1 Introduction

Convexity theory is an important foundation for studying a wide class of unrelated problems in a unified and general framework. Based on the notion of unique sharp minimizer, introduced by Polyak [11], and taking into account the works of Burke and Ferris [2], Patriksson [10], following Marcotte and Zhu [8], the variational inequalities have been strongly investigated by using the concept of weak sharp solution. We mention, in this respect, the works of Wu and Wu [15], Oveisiha and Zafarani [9], Alshahrani et al. [1], Liu and Wu [6] and Zhu [16].

In this paper, motivated and inspired by the ongoing research in this area, we introduce and investigate a new class of scalar variational inequalities. More precisely, by using several variational techniques presented in Clarke [3], Treanţă [12, 13] and Treanţă and Arana-Jiménez [14], we develop a new mathematical framework on controlled continuous-time variational inequalities governed by convex path-independent curvilinear integral functionals and, under some conditions and using a dual gap-type functional, we provide some characterization results for the associated solution set. As it is very well-known, the functionals of mechanical work type, due to their physical meaning, become very important in applications. Thus, the importance of this paper is supported both from theoretical and practical reasonings. As well, the ideas and techniques of this paper may stimulate further research in this dynamic field.

2 Notations, Working Hypotheses and Problem Formulation

In this paper, we will consider the following notations and working hypotheses:

\( \displaystyle \blacktriangleright \) two finite dimensional Euclidean spaces, \( \displaystyle \mathbb {R}^{n} \) and \( \displaystyle \mathbb {R}^{k} \);

\( \displaystyle \blacktriangleright \) \( \displaystyle \varTheta \subset \mathbb {R}^{m} \) is a compact domain in \( \displaystyle \mathbb {R}^{m} \) and the point \( \displaystyle \varTheta \ni t = (t^{\beta }), \, \beta = \overline{1,m} \), is a multi-parameter of evolution;

\( \displaystyle \blacktriangleright \) for \( \displaystyle t_{1} = (t_{1}^{1},\ldots ,t_{1}^{m}), \; t_{2} = (t_{2}^{1},\ldots ,t_{2}^{m}) \) two different points in \( \displaystyle \varTheta \), let \( \displaystyle \varTheta \supset \varUpsilon : t = t(\theta ), \theta \in [a,b] \) (or \( \displaystyle t \in \overline{t_{1},t_{2}} \)) be a piecewise smooth curve joining the points \( \displaystyle t_{1} \) and \( \displaystyle t_{2} \) in \( \displaystyle \varTheta \);

\( \displaystyle \blacktriangleright \) for \( \displaystyle \mathcal {U} \subseteq \mathbb {R}^{k}, \; \mathcal {P}:= \varTheta \times \mathbb {R}^{n} \times \mathcal {U} \) and \( \displaystyle i = \overline{1,n}, \; \beta = \overline{1,m}, \; \varsigma = \overline{1,q}\), we define the following continuously differentiable functions

$$ \displaystyle V = \left( V^{i}_{\beta }\right) : \mathcal {P} \rightarrow \mathbb {R}^{nm}, \quad W = \left( W_{\varsigma }\right) : \mathcal {P}\rightarrow \mathbb {R}^{q}; $$

\( \displaystyle \blacktriangleright \) for \( \displaystyle x_{\beta } := \frac{\partial x}{\partial t^{\beta }}, \; \beta = \overline{1,m} \), let \( \displaystyle \overline{\mathcal {\mathbf {X}}} \) be the space of piecewise smooth state functions \( \displaystyle x: \varTheta \rightarrow \mathbb {R}^{n} \) with the norm

$$ \displaystyle \parallel x \parallel = \parallel x \parallel _{\infty } + \sum _{\beta = 1}^{m}\parallel x_{\beta } \parallel _{\infty }, \quad \forall x \in \overline{\mathcal {\mathbf {X}}}; $$

\( \displaystyle \blacktriangleright \) also, denote by \( \displaystyle \overline{\mathcal {\mathbf {U}}} \) the space of piecewise continuous control functions \( \displaystyle u: \varTheta \rightarrow \mathcal {U} \), endowed with the uniform norm \( \displaystyle \parallel \cdot \parallel _{\infty } \);

\( \displaystyle \blacktriangleright \) consider \( \displaystyle \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \) equipped with the Euclidean inner product

$$ \displaystyle \langle (x,u); (y,w) \rangle = \int _{\varUpsilon }[x(t)\cdot y(t) + u(t)\cdot w(t)]dt^{\beta }, \quad \forall (x,u), (y,w) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} $$

and the induced norm;

\( \displaystyle \blacktriangleright \) denote by \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) a nonempty, closed and convex subset of \( \displaystyle \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \), defined as

$$ \displaystyle \mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}} = \{ (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} : \frac{\partial x^{i}}{\partial t^{\beta }} = V^{i}_{\beta }\left( t, x, u \right) , \; W \left( t, x, u \right) \le 0, $$
$$ \displaystyle x(t_{1}) = x_{1}, \; x(t_{2}) = x_{2} \}, $$

where \( \displaystyle x, u \) are the simplified notations for \( \displaystyle x(t), u(t) \) and \(x_{1}\) and \(x_{2}\) are given;

\( \displaystyle \blacktriangleright \) assume the continuously differentiable functions \( \displaystyle V_{\beta } = \left( V^{i}_{\beta }\right) , \; i = \overline{1,n}, \; \beta = \overline{1,m}, \) satisfy the following conditions of complete integrability

$$ \displaystyle D_{\zeta }V^{i}_{\beta } = D_{\beta }V^{i}_{\zeta }, \quad \beta , \zeta = \overline{1,m}, \; \beta \ne \zeta , \; i = \overline{1,n}, $$

where \( \displaystyle D_{\zeta } \) denotes the total derivative operator;

\( \displaystyle \blacktriangleright \) for any two q-tuples \( \displaystyle a = \left( a_{1}, ..., a_{q} \right) , b = \left( b_{1}, ..., b_{q} \right) \) in \( \displaystyle \mathbb {R}^{q} \), the following rules

$$ \displaystyle a = b \Leftrightarrow a_{\varsigma } = b_{\varsigma }, \quad a \le b \Leftrightarrow a_{\varsigma } \le b_{\varsigma }, $$
$$ \displaystyle a< b \Leftrightarrow a_{\varsigma } < b_{\varsigma }, \quad a \preceq b \Leftrightarrow a \le b, \; a \ne b, \quad \varsigma = \overline{1,q} $$

are assumed.

Note. Further, in this paper, it is assumed summation on repeated indices.

In the following, \( \displaystyle J^{1}(\mathbb {R}^{m}, \mathbb {R}^{n}) \) denotes the first-order jet bundle associated to \( \displaystyle \mathbb {R}^{m} \) and \( \displaystyle \mathbb {R}^{n} \). For \( \displaystyle \beta = \overline{1,m} \), we consider the real-valued continuously differentiable functions (closed Lagrange 1-form densities) \( \displaystyle l_{\beta }, s_{\beta }, r_{\beta }: J^{1}(\mathbb {R}^{m}, \mathbb {R}^{n}) \times \mathcal {U} \rightarrow \mathbb {R} \) and, for \( \displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \), define the following path-independent curvilinear integral functionals:

$$ \displaystyle L, S, R: \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \rightarrow \mathbb {R}, \quad L(x,u) = \int _{\varUpsilon } l_{\beta } \left( t, x, x_{\vartheta },u \right) dt^{\beta }, $$
$$ \displaystyle S(x,u) = \int _{\varUpsilon } s_{\beta } \left( t, x, x_{\vartheta },u \right) dt^{\beta }, \quad R(x,u) = \int _{\varUpsilon } r_{\beta } \left( t, x, x_{\vartheta },u \right) dt^{\beta }. $$

Definition 2.1

The scalar functional \( \displaystyle L(x,u) \) is called convex on \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) if, for any \( \displaystyle (x,u), (x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \), the following inequality

$$ \displaystyle L(x,u) - L(x^{0},u^{0}) $$
$$ \displaystyle \ge \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) (x - x^{0}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) D_{\vartheta }(x - x^{0}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) (u - u^{0})\right] dt^{\beta } $$

is satisfied.

Definition 2.2

For \( \displaystyle \beta = \overline{1,m} \), the variational (functional) derivative \( \displaystyle \delta _{\beta } L(x,u) \) of the scalar functional \( \displaystyle L: \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \rightarrow \mathbb {R}, \; L(x,u) = \int _{\varUpsilon } l_{\beta } \left( t, x, x_{\vartheta }, u \right) dt^{\beta } \), is defined as

$$ \displaystyle \delta _{\beta } L(x,u) = \frac{\delta _{\beta } L}{\delta x} + \frac{\delta _{\beta } L}{\delta u}, $$

with

$$ \displaystyle \frac{\delta _{\beta } L}{\delta x} = \frac{\partial l_{\beta }}{\partial x} \left( t, x, x_{\vartheta }, u \right) - D_{\vartheta }\frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x, x_{\vartheta }, u \right) \in \overline{\mathcal {\mathbf {X}}}, $$
$$ \displaystyle \frac{\delta _{\beta } L}{\delta u} = \frac{\partial l_{\beta }}{\partial u} \left( t, x, x_{\vartheta }, u \right) \in \overline{\mathcal {\mathbf {U}}} $$

and, for \( \displaystyle (\psi , \varPsi ) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \), with \( \displaystyle \psi (t_{1}) = \psi (t_{2}) = 0 \), the following relation

$$ \displaystyle \langle (\frac{\delta _{\beta } L}{\delta x}, \frac{\delta _{\beta } L}{\delta u}); (\psi , \varPsi ) \rangle = \int _{\varUpsilon }\left[ \frac{\delta _{\beta } L}{\delta x}(t)\cdot \psi (t) + \frac{\delta _{\beta } L}{\delta u}(t)\cdot \varPsi (t) \right] dt^{\beta } $$
$$ \displaystyle = \lim _{\varepsilon \rightarrow 0}\frac{L(x + \varepsilon \psi , u + \varepsilon \varPsi ) - L(x,u)}{\varepsilon } $$

is satisfied.

Working assumptions. (i) In this work, it is assumed that the inner product between the variational derivative of a scalar functional and an element \( \displaystyle (\psi , \varPsi ) \) in \( \displaystyle \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \) is accompanied by the condition \( \displaystyle \psi (t_{1}) = \psi (t_{2}) = 0 \).

(ii) Assume that

$$ \displaystyle dU := D_{\vartheta }\left[ \frac{\partial l_{\beta }}{\partial x_{\vartheta }} (x - x^{0}) \right] dt^{\beta } $$

is an exact total differential and satisfies \( \displaystyle U(t_{1}) = U(t_{2}) \).

At this point, we have the necessary mathematical tools to formulate the following controlled variational inequality problem: find \( \displaystyle (x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) such that

$$ \displaystyle \displaystyle (CVIP)\qquad \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) (x - x^{0}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) D_{\vartheta }(x - x^{0}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) (u - u^{0})\right] dt^{\beta } \ge 0, $$

for any \( \displaystyle (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \). The dual controlled variational inequality problem associated to (CVIP) is formulated as follows: find \( \displaystyle (x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) such that

$$ \displaystyle \displaystyle (DCVIP)\qquad \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x, x_{\vartheta }, u \right) (x - x^{0}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x, x_{\vartheta }, u \right) D_{\vartheta }(x - x^{0}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, x, x_{\vartheta }, u \right) (u - u^{0})\right] dt^{\beta } \ge 0, $$

for any \( \displaystyle (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \).

Denote by \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) and \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})_{*} \) the solution set associated to (CVIP) and (DCVIP), respectively, and assume they are nonempty.

Remark 2.1

As it can be easily seen (see (ii) in Working assumptions), we can reformulate the above controlled variational inequality problems as follows: find \( \displaystyle (x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) such that

$$ \displaystyle \displaystyle (CVIP)\qquad \langle (\frac{\delta _{\beta } L}{\delta x^{0}}, \frac{\delta _{\beta } L}{\delta u^{0}}); (x - x^{0}, u - u^{0}) \rangle \ge 0, \quad \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}, $$

respectively: find \( \displaystyle (x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) such that

$$ \displaystyle \displaystyle (DCVIP)\qquad \langle (\frac{\delta _{\beta } L}{\delta x}, \frac{\delta _{\beta } L}{\delta u}); (x - x^{0}, u - u^{0}) \rangle \ge 0, \quad \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}. $$

In the following, in order to describe the solution set \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) associated to (CVIP), we introduce the following gap-type path-independent curvilinear integral functionals.

Definition 2.3

For \( \displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}\), the primal gap-type path-independent curvilinear integral functional associated to (CVIP) is defined as

$$ \displaystyle S(x,u) = \max _{(x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}\{ \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x, x_{\vartheta }, u \right) (x - x^{0}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x, x_{\vartheta }, u \right) D_{\vartheta }(x - x^{0}) + \frac{\partial l_{\beta }}{\partial u} \left( t, x, x_{\vartheta }, u \right) (u - u^{0})\right] dt^{\beta } \}, $$

and the dual gap-type path-independent curvilinear integral functional associated to (CVIP) is defined as follows

$$ \displaystyle R(x,u) = \max _{(x^{0},u^{0}) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}} \{\int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) (x - x^{0}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) D_{\vartheta }(x - x^{0}) + \frac{\partial l_{\beta }}{\partial u} \left( t, x^{0}, x^{0}_{\vartheta }, u^{0} \right) (u - u^{0})\right] dt^{\beta } \}. $$

For \( \displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \), we introduce the following notations:

$$ \displaystyle \mathcal {A}(x,u) := \{(z,\nu ) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}: S(x,u) = \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x, x_{\vartheta }, u \right) (x - z) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x, x_{\vartheta }, u \right) D_{\vartheta }(x - z) + \frac{\partial l_{\beta }}{\partial u} \left( t, x, x_{\vartheta }, u \right) (u - \nu )\right] dt^{\beta }\}, $$
$$ \displaystyle \mathcal {Z}(x,u) := \{ (z,\nu ) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}: R(x,u) = \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, z, z_{\vartheta }, \nu \right) (x - z) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, z, z_{\vartheta }, \nu \right) D_{\vartheta }(x - z) + \frac{\partial l_{\beta }}{\partial u} \left( t, z, z_{\vartheta }, \nu \right) (u - \nu )\right] dt^{\beta }\}. $$

In the following, in accordance with Marcotte and Zhu [8], we introduce some central definitions.

Definition 2.4

The polar set \( \displaystyle (\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}})^{\circ } \) associated to \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) is defined as

$$ \displaystyle (\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}})^{\circ } = \left\{ (y, w) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}: \langle (y,w); (x,u) \rangle \le 0, \; \forall (x,u) \in \mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}} \right\} . $$

Definition 2.5

The normal cone to \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) at \( \displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \) is defined as

$$ \displaystyle N_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(x,u) = \{(y, w) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}: \langle (y,w), (z, \nu ) - (x,u) \rangle \le 0, $$
$$ \displaystyle \forall (z,\nu ) \in \mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}}\}, \quad (x,u) \in \mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}}, $$
$$ \displaystyle N_{\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}}}(x,u) = \emptyset , \quad (x,u) \not \in \mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}} $$

and the tangent cone to \( \displaystyle \mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}} \) at \( \displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \) is \( \displaystyle T_{\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}}}(x,u) = \left[ N_{\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}}}(x,u)\right] ^{\circ }\).

Remark 2.2

By using the definition of normal cone at \(\displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \), we observe the following: \( \displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \Longleftrightarrow \left( -\frac{\delta _{\beta } L}{\delta x^{*}}, -\frac{\delta _{\beta } L}{\delta u^{*}}\right) \in N_{\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}}}(x^{*},u^{*})\).

3 Preliminary Results

In this section, in order to formulate and prove the main results of the paper, several auxiliary propositions are established.

Proposition 3.1

Let the path-independent curvilinear integral functional \( \displaystyle L(x,u) \) be convex on \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \). Then:

(i) the following equality

$$ \displaystyle \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x^{2}, x^{2}_{\vartheta }, u^{2} \right) (x^{1} - x^{2}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x^{2}, x^{2}_{\vartheta }, u^{2} \right) D_{\vartheta }(x^{1} - x^{2}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, x^{2}, x^{2}_{\vartheta }, u^{2} \right) (u^{1} - u^{2})\right] dt^{\beta } = 0 $$

is fulfilled, for any \( \displaystyle (x^{1}, u^{1}), (x^{2}, u^{2}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \);

(ii) \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \subset (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})_{*} \).

Remark 3.1

The property of continuity for the variational derivative \( \displaystyle \delta _{\beta } L(x,u) \) implies \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})_{*} \subset (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}\). By Proposition 3.1, we conclude \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} = (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})_{*} \). Also, the solution set \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})_{*} \) associated to (DCVIP) is a convex set and, consequently, the solution set \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) associated to (CVIP) is a convex set.

Proposition 3.2

Let the path-independent curvilinear integral functional \( \displaystyle R(x,u) \) be differentiable on \( \displaystyle \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \). Then the following ineguality

$$ \displaystyle \langle (\frac{\delta _{\beta } R}{\delta x}, \frac{\delta _{\beta } R}{\delta u}); (v, \mu ) \rangle \ge \langle (\frac{\delta _{\beta } L}{\delta y}, \frac{\delta _{\beta } L}{\delta w}); (v, \mu ) \rangle $$

is satisfied, for any \( \displaystyle (x,u), (v,\mu ) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}, \;(y,w) \in \mathcal {Z}(x,u) \).

Proposition 3.3

Let the path-independent curvilinear integral functional \( \displaystyle R(x,u) \) be differentiable on \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) and the path-independent curvilinear integral functional \( \displaystyle L(x,u) \) be convex on \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \). Also, assume the following implication

$$ \displaystyle \langle (\frac{\delta _{\beta } R}{\delta x^{*}},\frac{\delta _{\beta } R}{\delta u^{*}}); (v,\mu ) \rangle \ge \langle (\frac{\delta _{\beta } L}{\delta z}, \frac{\delta _{\beta } L}{\delta \nu }); (v, \mu ) \rangle \Longrightarrow (\frac{\delta _{\beta } R}{\delta x^{*}},\frac{\delta _{\beta } R}{\delta u^{*}}) = (\frac{\delta _{\beta } L}{\delta z}, \frac{\delta _{\beta } L}{\delta \nu }) $$

is true, for any \( \displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}, \; (v,\mu ) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}, \;(z,\nu ) \in \mathcal {Z}(x^{*},u^{*}) \). Then \( \displaystyle \mathcal {Z}(x^{*},u^{*}) = (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}, \; \forall (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \).

4 Main Results

In this section, taking into account the preliminary results established in the previous section, we investigate weak sharp solutions for the considered controlled variational inequality governed by convex path-independent curvilinear integral functional. Concretely, following Marcotte and Zhu [8], in accordance with Ferris and Mangasarian [4], the weak sharpness property of \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) associated to (CVIP) is studied. In this regard, two characterization results are established.

Definition 4.1

The solution set \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) associated to (CVIP) is called weakly sharp if there exists \( \displaystyle \gamma > 0 \) such that

$$ \displaystyle \gamma B \subset \left( \frac{\delta _{\beta } L}{\delta x^{*}}, \frac{\delta _{\beta } L}{\delta u^{*}}\right) + \left[ T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(x^{*},u^{*}) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(x^{*},u^{*})\right] ^{\circ }, \quad \forall (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}, $$

(see \( \displaystyle int (Q) \) the interior of the set Q and B the open unit ball in \( \displaystyle \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \)), or, equivalently,

$$ \displaystyle \left( - \frac{\delta _{\beta } L}{\delta x^{*}}, - \frac{\delta _{\beta } L}{\delta u^{*}}\right) \in int\left( \bigcap _{(\overline{x},\overline{u}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}\left[ T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(\overline{x},\overline{u}) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(\overline{x},\overline{u})\right] ^{\circ }\right) , $$

for all \(\displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}\).

Lemma 4.1

There exists \( \displaystyle \gamma > 0 \) such that

$$\begin{aligned} \gamma B \subset \left( \frac{\delta _{\beta } L}{\delta y}, \frac{\delta _{\beta } L}{\delta w}\right) + \left[ T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(y,w) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(y,w)\right] ^{\circ }, \quad \forall (y,w) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \end{aligned}$$
(1)

if and only if

$$\begin{aligned} \langle (\frac{\delta _{\beta } L}{\delta y}, \frac{\delta _{\beta } L}{\delta w}); (z, \nu ) \rangle \ge \gamma \parallel (z,\nu ) \parallel , \quad \forall (z,\nu ) \in T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(y,w) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(y,w). \end{aligned}$$
(2)

The first characterization result of weak sharpness for \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is formulated in the following theorem.

Theorem 4.1

Let the path-independent curvilinear integral functional \( \displaystyle R(x,u) \) be differentiable on \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) and the path-independent curvilinear integral functional \( \displaystyle L(x,u) \) be convex on \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \). Also, assume that:

(a) the following implication

$$ \displaystyle \langle (\frac{\delta _{\beta } R}{\delta x^{*}}, \frac{\delta _{\beta } R}{\delta u^{*}}); (v, \mu )\rangle \ge \langle (\frac{\delta _{\beta } L}{\delta z}, \frac{\delta _{\beta } L}{\delta \nu }); (v,\mu ) \rangle \Longrightarrow \left( \frac{\delta _{\beta } R}{\delta x^{*}}, \frac{\delta _{\beta } R}{\delta u^{*}}\right) = \left( \frac{\delta _{\beta } L}{\delta z}, \frac{\delta _{\beta } L}{\delta \nu }\right) $$

is true, for any \( \displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}, \; (v,\mu ) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}, \; (z,\nu ) \in \mathcal {Z}(x^{*},u^{*}) \);

(b) \( \displaystyle \left( \frac{\delta _{\beta } L}{\delta x^{*}}, \frac{\delta _{\beta } L}{\delta u^{*}}\right) \) is constant on \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \).

Then \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is weakly sharp if and only if there exists \( \displaystyle \gamma > 0 \) such that

$$ \displaystyle R(x,u) \ge \gamma d\left( (x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}\right) , \quad \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}, $$

where \( \displaystyle d\left( (x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}\right) = \min _{(y,w) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}\parallel (x,u) - (y,w) \parallel \).

Proof

“\(\Longrightarrow \)” Consider \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is weakly sharp. Therefore, by Definition 4.1, there exists \( \displaystyle \gamma > 0 \) such that (1) (or (2)) is fulfilled. Further, taking into account the property of convexity for the solution set \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) associated to (CVIP) (see Remark 3.1), it follows \( \displaystyle proj_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(x,u) = (\hat{y}, \hat{w}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*},\; \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) and, following Hiriart-Urruty and Lemaréchal [5], we obtain \( \displaystyle (x,u) - (\hat{y}, \hat{w}) \in T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(\hat{y}, \hat{w}) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(\hat{y}, \hat{w}) \). By hypothesis and Lemma 4.1, we get

$$ \displaystyle \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, \hat{y}, \hat{y}_{\vartheta }, \hat{w} \right) (x - \hat{y}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, \hat{y}, \hat{y}_{\vartheta }, \hat{w} \right) D_{\vartheta }(x - \hat{y}) \right] dt^{\beta } $$
$$\begin{aligned} + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, \hat{y}, \hat{y}_{\vartheta }, \hat{w} \right) (u - \hat{w}) \right] dt^{\beta } \ge \gamma d((x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}), \quad \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}. \end{aligned}$$
(3)

Since

$$ \displaystyle R(x,u) \ge \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, \hat{y}, \hat{y}_{\vartheta }, \hat{w} \right) (x - \hat{y}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, \hat{y}, \hat{y}_{\vartheta }, \hat{w} \right) D_{\vartheta }(x - \hat{y}) \right] dt^{\beta } $$
$$ \displaystyle + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, \hat{y}, \hat{y}_{\vartheta }, \hat{w} \right) (u - \hat{w})\right] dt^{\beta }, \quad \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}, $$

by (3), we obtain \( \displaystyle R(x,u) \ge \gamma d((x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}), \; \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \).

“\(\Longleftarrow \)” Consider there exists \( \displaystyle \gamma > 0 \) such that \( \displaystyle R(x,u) \ge \gamma d((x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}), \; \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}. \) Obviously, for any \( \displaystyle (y,w) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \), the case \( \displaystyle T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(y,w) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(y,w) = \left\{ (0,0) \right\} \) involves \( \displaystyle \left[ T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(y,w) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(y,w)\right] ^{\circ } = \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \) and, consequently, \( \displaystyle \gamma B \subset \left( \frac{\delta _{\beta } L}{\delta y}, \frac{\delta _{\beta } L}{\delta w}\right) + \left[ T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(y,w) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(y,w)\right] ^{\circ }, \; \forall (y,w) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is trivial. In the following, let \( \displaystyle (0,0) \ne (\overline{x}, \overline{u}) \in T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(y,w) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(y,w) \) involving there exists a sequence \( \displaystyle (\overline{x}^{k}, \overline{u}^{k}) \) converging to \((\overline{x}, \overline{u})\) with \( \displaystyle (y,w) + t_{k}(\overline{x}^{k}, \overline{u}^{k})\in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \) (for some sequence of positive numbers \( \displaystyle \lbrace t_{k} \rbrace \) decreasing to zero), such that

$$ \displaystyle d((y,w) + t_{k}(\overline{x}^{k}, \overline{u}^{k}), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}) \ge d((y,w) + t_{k}(\overline{x}^{k}, \overline{u}^{k}), H_{\overline{x}, \overline{u}}) $$
$$\begin{aligned} = \frac{t_{k} \langle (\overline{x}, \overline{u}); (\overline{x}^{k}, \overline{u}^{k})\rangle }{\Vert (\overline{x}, \overline{u}) \Vert }, \end{aligned}$$
(4)

where \( \displaystyle H_{\overline{x}, \overline{u}} = \left\{ (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}:\langle (\overline{x}, \overline{u}); (x,u) - (y,w) \rangle = 0 \right\} \) is a hyperplane passing through (y, w) and orthogonal to \((\overline{x}, \overline{u})\). By hypothesis and (4), it results \( \displaystyle R((y,w) + t_{k}(\overline{x}^{k}, \overline{u}^{k})) \ge \gamma \frac{t_{k} \langle (\overline{x}, \overline{u}); (\overline{x}^{k}, \overline{u}^{k})\rangle }{\Vert (\overline{x}, \overline{u}) \Vert } \), or, equivalently (\( \displaystyle R(y,w) = 0, \; \forall (y,w) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}\)),

$$\begin{aligned} \frac{R((y,w) + t_{k}(\overline{x}^{k}, \overline{u}^{k})) - R(y,w)}{t_{k}} \ge \gamma \frac{\langle (\overline{x}, \overline{u}); (\overline{x}^{k}, \overline{u}^{k})\rangle }{\Vert (\overline{x}, \overline{u}) \Vert }. \end{aligned}$$
(5)

Further, by taking the limit for \( \displaystyle k \rightarrow \infty \) in (5) and using a classical result of functional analysis, we obtain

$$\begin{aligned} \lim _{\lambda \rightarrow 0}\frac{R((y,w) + \lambda (\overline{x}, \overline{u})) - R(y,w)}{\lambda } \ge \gamma \Vert (\overline{x}, \overline{u}) \Vert , \end{aligned}$$
(6)

where \( \displaystyle \lambda > 0 \). By Definition 2.2, the inequality (6) becomes

$$\begin{aligned} \langle (\frac{\delta _{\beta } R}{\delta y}, \frac{\delta _{\beta } R}{\delta w}); (\overline{x}, \overline{u}) \rangle \ge \gamma \Vert (\overline{x}, \overline{u}) \Vert . \end{aligned}$$
(7)

Now, taking into account the hypothesis and (7), for any \( \displaystyle (b,\upsilon ) \in B \), it follows \( \displaystyle \langle \gamma (b,\upsilon ) - (\frac{\delta _{\beta } L}{\delta y}, \frac{\delta _{\beta } L}{\delta w}); (\overline{x}, \overline{u}) \rangle = \langle \gamma (b,\upsilon ); (\overline{x}, \overline{u}) \rangle - \langle (\frac{\delta _{\beta } R}{\delta y},\frac{\delta _{\beta } R}{\delta w}); (\overline{x}, \overline{u}) \rangle \le \gamma \Vert (\overline{x}, \overline{u}) \Vert - \gamma \Vert (\overline{x}, \overline{u}) \Vert = 0 \) and the proof is complete.    \(\square \)

The second characterization result of weak sharpness for \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is based on the notion of minimum principle sufficiency property, introduced by Ferris and Mangasarian [4].

Definition 4.2

The controlled variational inequality (CVIP) satisfies minimum principle sufficiency property if \( \displaystyle \mathcal {A}(x^{*},u^{*}) = (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}, \; \forall (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \).

Lemma 4.2

The inclusion \( \displaystyle \arg \max _{(y,w) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}} \langle (x,u); (y,w) \rangle \subset (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is fulfilled for any \( \displaystyle (x,u) \in int\left( \bigcap _{(\overline{x}, \overline{u}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}\left[ T_{\mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}(\overline{x}, \overline{u}) \cap N_{(\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}}(\overline{x}, \overline{u})\right] ^{\circ }\right) \ne \emptyset \).

Theorem 4.2

Let the solution set \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) associated to (CVIP) be weakly sharp and the path-independent curvilinear integral functional \( \displaystyle L(x,u) \) be convex on \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \). Then (CVIP) satisfies minimum principle sufficiency property.

Theorem 4.3

Consider the functional \( \displaystyle R(x,u) \) is differentiable on \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) and the path-independent curvilinear integral functional \( \displaystyle L(x,u) \) is convex on \( \displaystyle \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \). Also, for any \( \displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}, \; (v,\mu ) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}}, \;(z,\nu ) \in \mathcal {Z}(x^{*},u^{*}) \), assume the following implication

$$ \displaystyle \langle (\frac{\delta _{\beta } R}{\delta x^{*}},\frac{\delta _{\beta } R}{\delta u^{*}}); (v,\mu ) \rangle \ge \langle (\frac{\delta _{\beta } L}{\delta z}, \frac{\delta _{\beta } L}{\delta \nu }); (v,\mu ) \rangle \Longrightarrow \left( \frac{\delta _{\beta } R}{\delta x^{*}},\frac{\delta _{\beta } R}{\delta u^{*}}\right) = \left( \frac{\delta _{\beta } L}{\delta z}, \frac{\delta _{\beta } L}{\delta \nu }\right) $$

is fulfilled and \( \displaystyle \left( \frac{\delta _{\beta } L}{\delta x^{*}}, \frac{\delta _{\beta } L}{\delta u^{*}}\right) \) is constant on \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \). Then (CVIP) satisfies minimum principle sufficiency property if and only if \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is weakly sharp.

Proof

“\(\Longrightarrow \)” Let (CVIP) satisfies minimum principle sufficiency property. In consequence, \( \displaystyle \mathcal {A}(x^{*},u^{*}) = (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \), for any \( \displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \). Obviously, for \( \displaystyle (x^{*},u^{*}) \in (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) and \( \displaystyle (x, u) \in \overline{\mathcal {\mathbf {X}}} \times \overline{\mathcal {\mathbf {U}}} \), we obtain

$$ \displaystyle R(x,u) \ge \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x^{*}, x^{*}_{\vartheta }, u^{*} \right) (x - x^{*}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x^{*}, x^{*}_{\vartheta }, u^{*} \right) D_{\vartheta }(x - x^{*}) \right] dt^{\beta } $$
$$\begin{aligned} + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, x^{*}, x^{*}_{\vartheta }, u^{*} \right) (u - u^{*})\right] dt^{\beta }. \end{aligned}$$
(8)

Further, for \( \displaystyle P(x,u) = \langle (\frac{\delta _{\beta } L}{\delta x^{*}},\frac{\delta _{\beta } L}{\delta u^{*}}); (x,u) \rangle , \; (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \), we get \( \displaystyle \mathcal {A}(x^{*},u^{*}) \) the solution set for \( \displaystyle \min _{(x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}}P(x,u) \). For other related investigations, the readers are directed to Mangasarian and Meyer [7]. We can write \( \displaystyle P(x,u) - P(\tilde{x},\tilde{u}) \ge \gamma d((x,u), \mathcal {A}(x^{*},u^{*})), \; \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}, \; (\tilde{x}, \tilde{u}) \in \mathcal {A}(x^{*},u^{*}) \), or, \( \displaystyle \langle (\frac{\delta _{\beta } L}{\delta x^{*}}, \frac{\delta _{\beta } L}{\delta u^{*}}); (x,u) - (x^{*},u^{*}) \rangle \ge \gamma d((x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}), \; \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}} \), or, equivalently,

$$ \displaystyle \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial x} \left( t, x^{*}, x^{*}_{\vartheta }, u^{*} \right) (x - x^{*}) + \frac{\partial l_{\beta }}{\partial x_{\vartheta }} \left( t, x^{*}, x^{*}_{\vartheta }, u^{*} \right) D_{\vartheta }(x - x^{*}) \right] dt^{\beta } $$
$$\begin{aligned} + \int _{\varUpsilon } \left[ \frac{\partial l_{\beta }}{\partial u} \left( t, x^{*}, x^{*}_{\vartheta }, u^{*} \right) (u - u^{*}) \right] dt^{\beta } \ge \gamma d((x,u), (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*}), \quad \forall (x,u) \in \mathcal {\mathbf {X}} \times \mathcal {\mathbf {U}}. \end{aligned}$$
(9)

By (8), (9) and Theorem 4.1, we get \( \displaystyle (\mathcal {\mathbf {X}}\times \mathcal {\mathbf {U}})^{*} \) is weakly sharp.

“\(\Longleftarrow \)” This is a consequence of Theorem 4.2.    \(\square \)