Regulation of Viable and Optimal Cohorts

Abstract

This study deals with the evolution of (scalar) attributes (resources or income in evolutionary demography or economics, position in traffic management, etc.) of a population of “mobiles” (economic agents, vehicles, etc.). The set of mobiles sharing the same attributes is regarded as an instantaneous cohort described by the number of its elements. The union of instantaneous cohorts during a mobile window between two attributes is a cohort. Given a measure defining the number of instantaneous cohorts, the accumulation of the mobile attributes on a evolving mobile window is the measure of the cohort on this temporal mobile window. Imposing accumulation constraints and departure conditions, this study is devoted to the regulation of the evolutions of the attributes which are

• 1. viable in the sense that the accumulations constraints are satisfied at each instant;

• 2. and, among them, optimal, in the sense that both the duration of the temporal mobile window is maximum and that the accumulation on this temporal mobile window is the largest viable one. This value is the “accumulation valuation” function.

Viable and optimal evolutions under accumulation constraints are regulated by an “implicit Volterra integro-differential inclusion” built from the accumulation valuation function, solution to an Hamilton–Jacobi–Bellman partial differential equation under constraints which is constructed for this purpose.

Appendix: Tangent Cones and Derivatives

The Viability Theorem²⁴ characterizes the viability of a subset under tangent cones: it states that, under adequate assumptions, a closed subset is viable under a differential inclusion if at every element, there exists a velocity “tangent” to the set at this element. Hence, the results of viability theory depend upon the definition of tangent to a closed subset.

By definition, the contingent cone $T_{K}(x)$ to $K$ at $x \in K$ is the set of directions $v \in X$ such that there exist a sequence $h_{n} \mapsto 0+$ from the right and a sequence $v_{n} \rightarrow v$ satisfying

$$\begin{aligned} \forall \; n \ge 0, \; \; x+h_{n}v_{n} \; \in \; K \end{aligned}$$

(112)

This statement assumes that the values $x+h_{n}v_{n}$ are forecast (but not known) in the future since $h_{n}>0$. This is the reason why they are called prospective contingent directions (defined for departure positions, for instance).

Since only the past can be known, it may be reasonable to look for directions $\overleftarrow{v}$ such that there exist a sequence $h_{n} \mapsto 0+$ from the right and a sequence $\overleftarrow{v}_{n} \rightarrow \overleftarrow{v}$ satisfying

$$\begin{aligned} \forall \; n \ge 0, \; \; x-h_{n}\overleftarrow{v}_{n} \; \in \; K \end{aligned}$$

(113)

This implies that $\overleftarrow{v} \in -T_{K}(x)=: \overleftarrow{T}_{K}(x)$. They are called retrospective contingent directions (defined for arrival positions, for instance).

We associate with the concept of tangent cone the concept of contingent derivative $DF(x,y)$ of a set-valued map $F:X \leadsto Y$ at a pair $(x,y) \in \mathrm{Graph}(F)$ of its graph, graphically defined by

$$\begin{aligned} \mathrm{Graph}(DF(x,y)) \; := \; T_{\mathrm{Graph}(F)}(x,y) \end{aligned}$$

(114)

This means that $v \in DF(x,y)$ if and only if there exist sequences $h_{n}\rightarrow 0+$, $u_{n} \rightarrow u$ and $v_{n} \rightarrow v$ such that

$$\begin{aligned} v_{n} \; \in \; \frac{F(x+h_{n}u_{n})-y}{h_{n} } \end{aligned}$$

(115)

In other words, the contingent derivative is the limit of “set-valued differential quotients”.

The retrospective contingent derivative $\overleftarrow{D}F(x,y)$ of a set-valued map $F:X \leadsto Y$ at a pair $(x,y) \in \mathrm{Graph}(F)$ of its graph is graphically defined by

$$\begin{aligned} \mathrm{Graph}(\overleftarrow{D} F(x,y)) \; := \; -T _{\mathrm{Graph}(F)}(x,y) \end{aligned}$$

(116)

This means that $\overleftarrow{v} \in \overleftarrow{D}F(x,y)(\overleftarrow{u})$ if and only if there exist sequences $h_{n}\rightarrow 0+$, $\overleftarrow{u}_{n} \rightarrow \overleftarrow{u}$ and $\overleftarrow{v}_{n} \rightarrow \overleftarrow{v}$ such that

$$\begin{aligned} \overleftarrow{v}_{n} \; \in \; \frac{y-F(x-h_{n}\overleftarrow{u}_{n})}{h_{n} } \end{aligned}$$

(117)

Extended functions $V: X \mapsto \{-\infty \} \cup \mathbb {R}\cup \{+\infty \}$ are characterized by their hypographs defined by

$$\begin{aligned} \mathcal{H}yp(V) \; := \; \left\{ (x,y) \in X \times \mathbb {R}\; \text{ such } \text{ that } \;y \; \le \; V(x) \right\} \end{aligned}$$

(118)

Extended functions $V: X \mapsto \{-\infty \} \cup \mathbb {R}\cup \{+\infty \}$ are characterized either by their epigraphs or hypographs defined respectively by

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal{E}p(V) \; := \; \left\{ (x,y) \in X \times \mathbb {R} \; \text{ such } \text{ that } \;y \; \ge \; V(x) \right\} \\ \mathcal{H}yp(V) \; := \; \left\{ (x,y) \in X \times \mathbb {R} \; \text{ such } \text{ that } \;y \; \le \; V(x) \right\} \\ \end{array} \right. \end{aligned}$$

(119)

We define epi and hypo derivatives $D_{\uparrow }V(x)$ and $D_{\downarrow } V(x)$ by

$$\begin{aligned} \mathcal{E}p(D_{\uparrow }V(x)) \; := \; T_{\mathcal{E}p(V)}(x,V(x)) \;\mathrm{and}\; \mathcal{H}yp(D_{\downarrow }V(x)) \; := \; T_{\mathcal{H}ypp(V)}(x,V(x)) \end{aligned}$$

(120)

and their retrospective epi and hypo derivatives $\overleftarrow{D}_{\uparrow }V(x)$ and $\overleftarrow{D}_{\downarrow } V(x)$ by

$$\begin{aligned} \mathcal{E}p(\overleftarrow{D}_{\uparrow }V(x)) := - T_{\mathcal{E}p(V)}(x,V(x)) \mathrm{and}\; \mathcal{H}yp(\overleftarrow{D}_{\downarrow }V(x)) := - T_{\mathcal{H}ypp(V)}(x,V(x)) \end{aligned}$$

(121)

One can check that

$$\begin{aligned} \overleftarrow{D}_{\downarrow } V(x)(\overleftarrow{u}) = - \overrightarrow{D}_{\uparrow } V(x)(-\overleftarrow{u}) \end{aligned}$$

(122)

We refer to Set-Valued Analysis, [18, Aubin and Frankowska], Traffic Networks as Information Systems. A viability Approach, [12, Aubin], and to [17, Aubin et al.] for more detailed consequences of these concepts.