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We prove the interior approximate controllability for the following 2&#x00D7;2 reaction-diffusion system with cross-diffusion matrix ut=a&#x0394;u&#x2212;&#x03B2;(&#x2212;&#x0394;)1/2u+b&#x0394;v+1&#x03C9;f1(t,x) in (0,&#x03C4;)&#x00D7;&#x03A9;, vt=c&#x0394;u&#x2212;d&#x0394;v&#x2212;&#x03B2;(&#x2212;&#x0394;)1/2v+1&#x03C9;f2(t,x) in (0,&#x03C4;)&#x00D7;&#x03A9;, u=v=0, on (0,T)&#x00D7;&#x2202;&#x03A9;, u(0,x)=u0(x), v(0,x)=v0(x), x&#x2208;&#x03A9;, where &#x03A9; is a bounded domain in &#x211D;N&#x2005;(N&#x2265;1), u0,v0&#x2208;L2(&#x03A9;), the 2&#x00D7;2 diffusion matrix D=[abcd] has semisimple and positive eigenvalues 0&#x003C;&#x03C1;1&#x2264;&#x03C1;2, &#x03B2; is an arbitrary constant, &#x03C9; is an open nonempty subset of &#x03A9;, 1&#x03C9; denotes the characteristic function of the set &#x03C9;, and the distributed controls f1,f2&#x2208;L2([0,&#x03C4;];L2(&#x03A9;)). Specifically, we prove the following statement: if &#x03BB;11/2&#x03C1;1+&#x03B2;&#x003E;0 (where &#x03BB;1 is the first eigenvalue of &#x2212;&#x0394;), then for all &#x03C4;&#x003E;0 and all open nonempty subset &#x03C9; of &#x03A9; the system is approximately controllable on [0,&#x03C4;]

Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix