Observability of Systems under Uncertainty

The evolution of the state $x( \cdot )$ of a system under uncertainty governed by a differential inclusion \[{\text{for almost all }}t \in [0,T],\qquad x'(t) \in F(t,x(t))\] is observed through an observation map H: \[\forall t \in [0,T],\quad y(t) \in H(x(t)).\] The set-valued character due to the uncertainty leads to the introduction of the following: Sharp input-output map, which is the (usual) product \[\forall x_0 \in X,\quad I\_(x_0 ): = (H \circ \mathcal{S})(x_0 ): = \mathop \cup \limits_{x( \cdot ) \in \mathcal{S}(x_0 )} H(x( \cdot )).\] Hazy input-output map, which is the square product\[\forall x_0 \in X,\quad I_ + (x_0 ): = (H\square \mathcal{S})(x_0 ): = \mathop \cap \limits_{x( \cdot ) \in \mathcal{S}(x_0 )} H(x( \cdot )).\] Where $\mathcal{S}$ denotes the solution map, recovering the input $x_0 $ from the outputs $I_ - x_0 $ or $I_ + x_0 $ means that these input–output maps are “injective” in the sense that, locally, \[x_1 \ne x_2 \Rightarrow I(x_1 ) \cap I(x_2 ) = \emptyset.\] Criteria for both sharp and hazy local observability are provided in terms of (global sharp and hazy observability of the variational inclusion \[w'(t) \in DF(t,\bar x(t),\bar x'(t))(w(t)),\] which is a “linearization” of the differential inclusion along a solution $\bar x( \cdot )$, where for almost all t, $DF(t,x,y)(u)$ denotes the contingent derivative of the set-valued map $F(t, \cdot , \cdot )$ at a point $(x,y)$ of its graph. These conclusions are reached by implementing the following strategy: 1. Provide a general principle of local injectivity and observability of a set-valued map I, which derives these properties from the fact that the kernel of an adequate derivative of I is equal to zero. 2. Supply chain rule formulas that allow computation of the derivatives of the usual product $I_ - $ and the square product $I_ + $ from the derivatives of the observation map H and the solution map $\mathcal{S}$. Characterize the various derivatives of the solution map $\mathcal{S}$ in terms of the solution maps of the associated variational inclusions. 4. Piece together these results for deriving local sharp and hazy observability of the original system from sharp and hazy observability of the variational inclusions. 5. Study global sharp and hazy observability of the variational inclusions