If you are not familiar with wsadmin.sh (Linux) / wsadmin.bat (Windows), check out my article Getting Started with wsadmin.

Here is an example of how to list applications using wsadmin.

~]$ /opt/WebSphere/AppServer/profiles/your_profile/bin/wsadmin.sh -lang jython -c "print AdminApp.list()"
WASX7209I: Connected to process "dmgr" on node Node01 using SOAP connector;  The type of process is: DeploymentManager
myApp01EAR
myApp02EAR
myApp03EAR

And then the following wsadmin command can be used to determine if an application is running. If no output is returned, the application is not running. If output is returned, the application is running.

~]$ /opt/WebSphere/AppServer/profiles/your_profile/bin/wsadmin.sh -lang jython -c "print AdminControl.completeObjectName('type=Application,name=myApp01EAR,*')"
WebSphere:name=myApp01EAR,process=Cluster01,platform=dynamicproxy,node=Node01,J2EEName=myApp01EAR,Server=AppSrv01,version=8.5.5.21,type=Application,mbeanIdentifier=cells/Cell01/applications/myApp01EAR.ear/deployments/myApp01EAR/deployment.xml#ApplicationDeployment_1403144753115,cell=Cell01,spec=1.0