Common use of Main Result Clause in Contracts

Main Result. Time-varying Communication Topology the communication graph. Then (9) yields min . V˙ ≤ −λ .BT BΣ |x¯|2 + |x¯| B T B δ √ uΣ m In this section we treat the case when the communication ǁ ǁ topology is time-varying, allowing each agent to lose/create ≤ −λmin .BT BΣ |x¯| |x¯| − BT B δu√m λmin(BT B) new communication links with other agents as the closed- loop system evolves. The problem in this case is that it’s Thus, all solutions of the closed-loop system enter the ball not possible to use V = 1 x¯T x¯ as a common Lyapunov . BT Σ √ B δu m function for the switched system, since the vector x¯ changes discontinuously whenever edges are added or deleted when x : |x¯| ≤ λmin (BT B) the communication topology changes. A different energy function is used and in particular, the function λmin(BT B) centered at x¯ = 0 of radius ǁB Bǁδu m in finite time. In the case of a logarithmic quantizer we have q = ql and |ql (x¯) − x¯| ≤ δl |x¯| and (9) yields W = max {x1, . . . , xN } − min {x1, . . . , xN } (16) which can act as a common Lyapunov function for the ∆ V˙ ≤ −λmin .BT BΣ |x¯| + BT B δl |x¯|2 , switched system. ∆ Let xmax = max {x1, . . . , xN } , xmin = so that V˙ ≤ − |x¯|

Main Result. Time-varying Communication Topology ¨B ¨ u m the communication graph. Then (9) yields min . V˙ ≤ −λ .BT BΣ |x¯|2 −λmin BT B | x¯|2 + |x¯| B ¨ T B B¨ δ √ uΣ m In this section we treat the case when the communication ǁ ǁ topology is time-varying, allowing each agent to lose/create ≤ −λmin .BT BΣ BT B |x¯| |x¯| − BT B δu√m λmin(BT B) new communication links with other agents as the closed- loop system evolves. The problem in this case is that it’s Thus, all solutions of the closed-loop system enter the ball not possible to use V = 1 x¯T x¯ as a common Lyapunov . BT Σ ( ¨ ¨BT ¨ ) √ B B¨ δu m function for the switched system, since the vector x¯ changes discontinuously whenever edges are added or deleted when x : |x¯| ≤ λmin (BT B) the communication topology changes. A different energy function is used and in particular, the function λmin(BT B) centered at x¯ = 0 of radius ǁB Bǁδu m B δu in finite time. λmin(BT B) In the case of a logarithmic quantizer we have q = ql and |ql (x¯) − x¯| ≤ δl |x¯| and (9) yields W = max {x1, . . . , xN } − min {x1, . . . , xN } (16) which can act as a common Lyapunov function for the ∆ V˙ ≤ −λmin .BT BΣ |x¯| + BT B |x¯| ¨ + ¨BT ¨ ∆ B¨ δl |x¯|2 , switched system. ∆ Let xmax = max {x1, . . . , xN } , xmin = so that V˙ ≤ − |x¯||x¯| λmin BT B ¨ − ¨BT ¨ B¨ δl