MathDB

A cat is trying to catch a mouse in the non-negative quadrant

N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.

At time

t=0

the cat is at

(1,1)

and the mouse is at

(0,0)

. The cat moves with speed

\sqrt{2}

such that the position

c(t)=(c_1(t),c_2(t))

is continuous, and differentiable except at finitely many points; while the mouse moves with speed

1

such that its position

m(t)=(m_1(t),m_2(t))

is also continuous, and differentiable except at finitely many points. Thus

c(0)=(1,1)

and

m(0)=(0,0)

;

c(t)

and

m(t)

are continuous functions of

t

such that

c(t),m(t)\in N

for all

t\geq 0

; the derivatives

c'(t)=(c'_1(t),c'_2(t))

and

m'(t)=(m'_1(t),m'_2(t))

each exist for all but finitely many

t

and

(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,

whenever the respective derivative exists.
At each time

t

the cat knows both the mouse's position

m(t)

and velocity

m'(t)

. Show that, no matter how the mouse moves, the cat can catch it by time

t=1

; that is, show that the cat can move such that

c(\tau)=m(\tau)

for some

\tau\in[0,1]

Problem Statement